Re: Question about PA and 1p
On 17 Jan 2012, at 20:06, Stephen P. King wrote:
On 1/17/2012 5:08 AM, Bruno Marchal wrote:
SNIP
- I disagree that set theory might be more primitive than
arithmetic. Why? First because arithmetic has been proved more
primitive than set theory, and less primitive than logic. With
logic we cannot define numbers. with set, we can define numbers,
even all of them (N, Z, Q, ... octonions, etc.). The natural
numbers are often defined by the von Neuman finite ordinal:
0 = { }
1 = {{{}}} = {0}
2 = {{}, {{}} } = {0, 1}
3 = {{}, {{}},{{}}},,{{} ,{{}} } } = {0, 1, 2}
...
n = {0, 1, 2, ..., n-1}
etc.
And you can define addition by the disjoint union cardinal, and
multiplication by the cardinal of cartesian product,
and then, you can *prove* the laws of addition and the laws of
multiplication. With arithmetic you cannot recover any axioms of
set theory, except for the hereditarily finite sets.
I am confused. It seems to me that you are admitting that sets
are more primitive than Arithmetic since what you wrote here is a
demonstration of how numbers supervene on set theoretic operations.
The fact that we can define the natural numbers via the von Neuman
finite ordinals is the equivalent of claiming that the natural
numbers emerge from the von Neuman finite ordinals (up to
isomorphism!), so I am confused by what you are claiming here! But
whichever is the most primitive, it is not more primitive than the
neutral foundation of existence in itself.
I just meant that sets are more complicated that natural numbers, so
by assuming sets you assume more than by assuming just the natural
numbers. With comp we have that assuming arithmetic is enough. Sets,
real numbers and the physical world are recovered in the epistemology
of relative natural numbers (that is a number + a universal numbers).
I have no clue what you mean by neutral foundation of existence in
itself.
I have another problem with set theories. There is no clear standard
model. For arithmetic there is. The set of Gödel number of true
arithmetical sentences is a highly complex set, but it is still well
defined. That is not the case for the set of Gödel numbers of set
theoretical sentences. I cannot be realist about sets.
Yet another problem: in the quantified self-reference logic on set
theory, B(P(x)), or [ ] P(x) has no easy meaning, and suffer from all
the Quine-Marcus critics of quantified model logic, where on the
contrary in the quantified self-reference logic on arithmetic BP(x) is
crystal clear, and defeat all those critics, by showing transparent
counter-example.
I will confess you, Stephen, that I have never really believe in Set
Theories. Set theories looks just like quite imaginative Löbian
numbers to me. But I know well ZF, and appreciate it as an interesting
logical object.
- As I said, I don't take the word Existence as a theory. I have
no clue what you mean by that. I was asking for a theory. You say
that by taking (N, +, *) as a primitive structure, I am no more
neutral monist, due to the use of + and *. This is not correct. It
would make neutral monism empty. We alway need ontological terms
(here 0, s(0) etc.) and laws relating those terms (here addition
and multiplication).
No, I am not making Existence as a theory, it is merely a
postulate of my overall theory
Theories are made of postulates. I don't see a postulate. Existence of
what?
(if you can call what I have been discussing a theory). I am using
the notion of Existence as it is defined in Objectivist
Epistemology. For example, as explained in this video lecture: http://www.youtube.com/watch?v=GfOS7xfxezAfeature=player_embedded
Neutral monist takes is empty in the sense that it shows the
coherent implication that the most basic ontological level cannot be
considered to have some definite set of properties to the exclusion
of others.
This does not make sense for me. Sorry. (I am not a philosopher). You
might have to elaborate.
Something primitive without any property cannot explain anything. That
is why physicist postulate particles and forces, and mathematicians
postulate numbers and laws or operations, or set and belonging
relations.
- All you arguments with the term physical are going through in
arithmetic, given that you agree that physical is not primitive.
For example, the physical world is not required to make sense of
what is a universal machine. It is required for human chatting on
the net, but such a physical world is provided by arithmetic.
Including concurrency.
WE simply might have to agree to disagree.
You cannot disagree with a theorem in a theory. You have to find a
flaw in the proof, or you have to disagree with the premise. If you
disagree with what I say above, I take it that you say no to the
digitalist doctor, and defend a non computationalist theory of mind.
- I don't do philosophy.
Re: Question about PA and 1p
On 16 Jan 2012, at 20:42, David Nyman wrote: On 16 January 2012 18:08, Bruno Marchal marc...@ulb.ac.be wrote: I do not need an extra God or observer of arithmetical truth, to interpret some number relation as computations, because the numbers, relatively to each other, already do that task. From their view, to believe that we need some extra-interpreter, would be like to believe that if your own brain is not observed by someone, it would not be conscious. I'm unclear from the above - and indeed from the rest of your comments - whether you are defining interpretation in a purely 3p way, or whether you are implicitly placing it in a 1-p framework - e.g. where you say above From their view. If you do indeed assume that numbers can have such views, then I see why you would say that they interpret themselves, because adopting the 1p view is already to invoke a kind of emergence of number-epistemology. But such an emergence is still only a manner of speaking from OUR point of view, in that I can rephrase what you say above thus: From their view, to believe that THEY need some extra-interpreter... without taking such a point of view in any literal sense. Are you saying that consciousness somehow elevates number-epistemology into strong emergence, such that their point of view and self-interpretation become indistinguishable from my own? It seems to me that this follows from UDA1-8. If not, then arithmetic if full of immaterial zombies, given that those computations does exist in arithmetic, in the usual sense of 17 is prime independently of me. Or you need to reify matter to singularize consciousness, but this is shown by the movie graph (UDA-8, MGA) to be a red herring type of move. Number relations does implement computations, in the same sense that brains' physics implement computations, by MGA. Now the 1p are related, not on any particular computations in the UD (or in arithmetic), but to all of them, making both matter and consciousness not Turing emulable, but still recoverable from the entire work of the UD (UD*) or from the whole arithmetical truth. The point of view of some numbers will not differ from yours, given that yours is given by infinitely many such numbers relations. OK? Bruno Bruno David On 16 Jan 2012, at 15:32, David Nyman wrote: On 16 January 2012 10:04, Bruno Marchal marc...@ulb.ac.be wrote: Actually you can define computation, even universal machine, by using only addition and multiplication. So universal machine exists in elementary arithmetic in the same sense as in the existence of prime number. That may be, but we were discussing interpretation. As you say above: YOU can define computation, even universal machine, by using only addition and multiplication (my emphasis). Not just ME. A tiny part of arithmetic can too. All universal numbers can do that. No need of first person notion. All this can be shown in a 3p way. Indeed, in arithmetic. Even without the induction axioms, so that we don't need Löbian machine. The existence of the UD for example, is a theorem of (Robinson) arithmetic. Now, that kinds of truth are rather long and tedious to show. This was shown mainly by Gödel in his 1931 paper (for rich Löbian theories). It is called arithmetization of meta-mathematics. I will try to explain the salt of it without being too much technical below. But this is surely, as you are wont to say, too quick. Firstly, in what sense can numbers in simple arithmetical relation define THEMSELVES as computation, or indeed as anything else than what they simply are? Here you ask a more difficult question. Nevertheless it admits a positive answer. I think that the ascription of self-interpretation to a bare ontology is superficial; it conceals an implicit supplementary appeal to epistemology, and indeed to a self. But can define a notion of 3-self in arithmetic. Then to get the 1- self, we go at the meta-level and combine it with the notion of arithmetical truth. That notion is NOT definable in arithmetic, but that is a good thing, because it will explain why the notion of first person, and of consciousness, will not be definable by machine. Hence it appears that some perspectival union of epistemology and ontology is a prerequisite of interpretation. OK. But the whole force of comp comes from the fact that you can define a big part of that epistemology using only the elementary ontology. Let us agree on what we mean by defining something in arithmetic (or in the arithmetical language). The arithmetical language is the first order (predicate) logic with equality(=), so that it has the usual logical connectives (, V, - , ~ (and, or, implies, not), and the quantifiers E and A, (it exists and for all), together with the special arithmetical symbols 0, s + and *. To illustrate an arithmetical definition, let me give you some definitions of simple concepts. We can
Re: Question about PA and 1p
On 17 January 2012 14:51, Bruno Marchal marc...@ulb.ac.be wrote: I think we are very close. And very close to Schroedinger intuition indeed. I think we are. However, I'm still uncomfortable about the single glance. I can see how one can talk about points of view in a 3p sense by, in effect, pointing to 3p entities and attributing 1p views to them. However, as soon as one actually *adopts* the 1p stance, one becomes restricted to what we experience as a *succession* of personally-selected instances, mutually-exclusive of all other such instances. It is tempting to go on thinking of this serialisation of 1p-as-experienced instances as though it was just the natural outcome of their continuing to co-exist all together, as in the 3p situation. But the trouble then is the lack of a rationale for recovering just this instance NOW, whilst simultaneously retaining the credible belief in its *substitution* by other such moments. To put it another way, a single uniquely-experienced point of view can't intelligibly be both somewhere and everywhere. Do you see my difficulty? David On 17 Jan 2012, at 13:51, David Nyman wrote: On 17 January 2012 09:43, Bruno Marchal marc...@ulb.ac.be wrote: Now the 1p are related, not on any particular computations in the UD (or in arithmetic), but to all of them, making both matter and consciousness not Turing emulable, but still recoverable from the entire work of the UD (UD*) or from the whole arithmetical truth. The point of view of some numbers will not differ from yours, given that yours is given by infinitely many such numbers relations. OK? I think so. Here's what we seem to be saying, in brief: 1) Start with the presupposition that consciousness supervenes on the point of view of a digital machine (i.e. CTM). OK (with some nuances, due to the fact that we don't know and eventually cannot know which machine we are, so that there are some difficulties needed to be met in relation with the notion of comp substitution level, usually implicit in most version of CTM). 2) Demonstrate how such machinery can logically encapsulate a point of view. OK. At least in the 3p sense. That's a consequence of Gödel's construction (arithmetization), mainly. 3) Argue that an infinity of such machinery emerges from arithmetic as a consequence of UD*. Yes. That is the first person global indeterminacy. The one you face in case a UD is running integrally in the physical universe, or the one you face if you accept that elementary arithmetic is independent of you (by MGA). 4) Show that 2) and 3) therefore entails an infinity of such points of view. Yes. But note the ambiguity. Here it means that we have to take into account all the non distinguishable (identical) 3p views appearing in arithmetic (or UD*). They will define the lasting, persisting, 1p observation. 5) Show that the conjunction of I am conscious now and assumptions 1), 2), 3) and 4) entails that consciousness supervenes on an infinity of points of view. OK. Note that without MGA, we could still believe that some primary physical universe is needed, but in a robust universe the reversal is already proved. From the above, given that I am conscious in the present moment, my current state is computationally entangled with other states comprising the memoires of DN, and is associated in a weaker sense, by 5), to all other such memoires. Yes. This seems to give us something like Schrödinger's association of consciousness with the whole that cannot be surveyed in a single glance. That might be possible. What might be possible is that such a consciousness is the atemporal consciousness of the universal person, which is the UM, or the LUM (for strong self-consciousness). Basically a LUM is a UM with a rich self (the UM have already a self, but cannot prove a lot about it). Of course, this selfsame narrowing of attention - i.e. the temporal, and temporary, isolation of one mutually-exclusive moment - is one of the givens and hence transcends the explanation. But it seems clear to me already be explained by the 3p self-description (by Bp, that is Gödel's provability predicate). What is really transcendent is nothing but the whole (arithmetical) truth, which provably transcend PA. Now we are richer than PA, and this concerns thus a non definable transcendental notion of truth. But it is only by means of such interpretative glances that number-epistemology can be elevated into the strong emergence of personal knowledge. Hmm... OK. But such interpretative glance is not much more than what you need to believe to accept that the excluded middle principle is correct for the intuitive arithmetical propositions (such an intuition is transcendental, but fully formalized, at the meta-level, for any *correct* machine, by Bp p. The p is definable by a LUM for a simpler LUM known to be correct by the first one. It is transcendental
Re: Question about PA and 1p
On 1/17/2012 5:08 AM, Bruno Marchal wrote:
SNIP
- I disagree that set theory might be more primitive than arithmetic.
Why? First because arithmetic has been proved more primitive than set
theory, and less primitive than logic. With logic we cannot define
numbers. with set, we can define numbers, even all of them (N, Z, Q,
... octonions, etc.). The natural numbers are often defined by the von
Neuman finite ordinal:
0 = { }
1 = {{{}}} = {0}
2 = {{}, {{}} } = {0, 1}
3 = {{}, {{}},{{}}},,{{} ,{{}} } } = {0, 1, 2}
...
n = {0, 1, 2, ..., n-1}
etc.
And you can define addition by the disjoint union cardinal, and
multiplication by the cardinal of cartesian product,
and then, you can *prove* the laws of addition and the laws of
multiplication. With arithmetic you cannot recover any axioms of set
theory, except for the hereditarily finite sets.
I am confused. It seems to me that you are admitting that sets are
more primitive than Arithmetic since what you wrote here is a
demonstration of how numbers supervene on set theoretic operations. The
fact that we can define the natural numbers via the von Neuman finite
ordinals is the equivalent of claiming that the natural numbers emerge
from the von Neuman finite ordinals (up to isomorphism!), so I am
confused by what you are claiming here! But whichever is the most
primitive, it is not more primitive than the neutral foundation of
existence in itself.
- As I said, I don't take the word Existence as a theory. I have no
clue what you mean by that. I was asking for a theory. You say that by
taking (N, +, *) as a primitive structure, I am no more neutral
monist, due to the use of + and *. This is not correct. It would make
neutral monism empty. We alway need ontological terms (here 0, s(0)
etc.) and laws relating those terms (here addition and multiplication).
No, I am not making Existence as a theory, it is merely a postulate
of my overall theory (if you can call what I have been discussing a
theory). I am using the notion ofExistence
http://books.google.com/books?id=VttF6CuC-cQCpg=PT170lpg=PT170dq=existence+Objectivist+epistemologysource=blots=d2ZAMFVpJMsig=CLaMS0Y9kVnB6UfbgwUsuCG3wsUhl=ensa=Xei=vsMVT7WNM8nWtgfEvaTRAwsqi=2ved=0CFUQ6AEwBg#v=onepageq=existence%20Objectivist%20epistemologyf=false
as it is defined in Objectivist Epistemology. For example, as explained
in this video lecture:
http://www.youtube.com/watch?v=GfOS7xfxezAfeature=player_embedded
Neutral monist takes is empty in the sense that it shows the
coherent implication that the most basic ontological level cannot be
considered to have some definite set of properties to the exclusion of
others.
- All you arguments with the term physical are going through in
arithmetic, given that you agree that physical is not primitive. For
example, the physical world is not required to make sense of what is a
universal machine. It is required for human chatting on the net, but
such a physical world is provided by arithmetic. Including concurrency.
WE simply might have to agree to disagree.
- I don't do philosophy. I offer you a technical result only. I still
don't know if you grasped it, or if you have any problem with it.
You result has deep philosophical implications and as a student of
philosophy I am very interested in it.
If you agree to assume that the brain works like a material machine,
then arithmetic is enough and more than arithmetic is necessarily
useless: it can only make the mind body problem unecessarily more
complex. Primitive matter (time, space) becomes like invisible horse.
Not epiphenomena, but epinomena.
Again, we have to agree to disagree on this. The necessity of
physical implementation cannot be dismissed otherwise the scientific
method itself is empty and useless. Without the definiteness that the
physical world offers us is accepted there can be only idle speculation,
we saw this kind of thinking in the Scholastics
http://en.wikipedia.org/wiki/Scholasticism and know well how that was
such a terrible waste of time. So why are you advocating a return to
that? Ideal monism was pushed hard by Bishop Berkeley and failed back in
the 18th century, its flaw - that the material world becomes causally
ineffective epiphenomena - is not solved by your result, it is only more
explicitly shown. You seem to think that it is a virtue. No, sorry, it
is not a virtue for the simple reason that it makes the falsification of
the theory impossible thus rendering it useless as an explanation.
My alternative hypothesis has the chance of being falsified as it
predicts that the physical worlds actually observe must be representable
as Boolean algebras (up to isomorphisms).
Onward!
Stephen
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Re: Question about PA and 1p
On 17 January 2012 20:02, Bruno Marchal marc...@ulb.ac.be wrote: The consciousness is the same, but with different input bits. It *is* the relative indeterminacy, intrinsic to all computational state rich enough to code a relatively universal number. The question why here and now is a constant on all such computational states, except in some amnesic, dissociated or contemplative mode, or in different attention mode. OK? snip Please tell me if this helped, or if I miss something. Or if you would dislike comp for this, that happens, we are near to the understanding that personal identity is an illusion, somehow, and many dislike this. Bruno, I have excerpted above the two pieces of your reply that most make me think that we do agree on much, but haven't yet quite succeeded in communicating the nuances of that agreement. I've certainly had the intuition, for much of my life, that personal identity is an illusion in some sense, as my story about why am I me and not you? was meant to convey. So this is not really the issue. I think perhaps that you are much more accustomed than I am to thinking in mathematical and formal-logical terms, and hence there can be some mutual difficulty in reconciling this point-for-point with my perhaps more analogical style of thinking and expression. My analogy to what you say is something like this. In terms of the comp framework, the present moment of DN equates to the selective supervenience of consciousness on some non-distinguishable sub-class of all possible computational states of the general class we customarily call observer moments. Such a primary selectivity immediately results in the association of this moment (through computational entanglement) with the class of states constituting all possible memoires of DN. It is this secondary association that assembles a personal identity distinguishable as DN. Consequently I see the primary action of the supervenience relation as selective of personal identity, but not wholly constitutive of it. The constitutive dimension seems to me to lie in the entanglements discoverable through that primary selectivity. I realise every time I think in this way that - especially, perhaps, for you Bruno - it might appear to imply a sort of solipsistic heresy. It seems to say that something-or-other is acting on behalf of everyone, but not at the same time. The possibility of the simultaneity of two mutually-exclusive conscious moments seems to be excluded. Of course, such mutual exclusion is quite consistent with - indeed essential to - the memoire of DN, whose experiential coherence is predicated on it. And indeed, considered separately, it is similarly consistent with other personal memoires, whose proper entanglements distinguish them from each other and from that of DN. If there is indeed a universal mind, it is one whose selective attention precludes for one moment ALL other moments, quite irrespective of the secondary association of that moment with some memoire or other. I've been thinking about this quite a lot recently. I realise it is only an analogy, and hopefully I may already have said enough for you to comment and perhaps point out where I may be pushing analogy too far or being naively literal. However, these recent thoughts have made me think again about why we might wrongly suppose our environment to be populated by zombies (i.e. persons whose conscious states can't be observed directly here-and-now) and even possibly a different way of thinking about the significance of MGA-type arguments (different for me, that is). However, I'll stop now, because I find that this particular moment in the memoire of DN is a really weary one! David On 17 Jan 2012, at 17:58, David Nyman wrote: On 17 January 2012 14:51, Bruno Marchal marc...@ulb.ac.be wrote: I think we are very close. And very close to Schroedinger intuition indeed. I think we are. However, I'm still uncomfortable about the single glance. I can see how one can talk about points of view in a 3p sense by, in effect, pointing to 3p entities and attributing 1p views to them. OK. However, as soon as one actually *adopts* the 1p stance, one becomes restricted to what we experience as a *succession* of personally-selected instances, mutually-exclusive of all other such instances. Not necessarily, but usually yes (I call that the struggle of life). In more contemplative mode, the emphasis on succession can drop. Computer-science theoretically, it is an indexical like I am the one in M. Observation select relative branch, and information bits are created. But relatively to the universal numbers in the neighborhood, that indexical, assuming comp, is entirely represented by one computational state. If that one is frozen, and put on some disk, then the future 1-p, relatively to that frozen state will depend on all histories going as near as possible to that state of affair, where it will make sense for being a
Re: Question about PA and 1p
On 14 Jan 2012, at 18:51, David Nyman wrote: On 14 January 2012 16:50, Stephen P. King stephe...@charter.net wrote: The problem is that mathematics cannot represent matter other than by invariance with respect to time, etc. absent an interpreter. Sure, but do you mean to say that the interpreter must be physical? I don't see why. And yet, as you say, the need for interpretation is unavoidable. Now, my understanding of Bruno, after some fairly close questioning (which may still leave me confused, of course) is that the elements of his arithmetical ontology are strictly limited to numbers (or their equivalent) + addition and multiplication. This emerged during discussion of macroscopic compositional principles implicit in the interpretation of micro-physical schemas; principles which are rarely understood as being epistemological in nature. Hence, strictly speaking, even the ascription of the notion of computation to arrangements of these bare arithmetical elements assumes further compositional principles and therefore appeals to some supplementary epistemological interpretation. In other words, any bare ontological schema, uninterpreted, is unable, from its own unsupplemented resources, to actualise whatever higher-level emergents may be implicit within it. But what else could deliver that interpretation/actualisation? What could embody the collapse of ontology and epistemology into a single actuality? Could it be that interpretation is finally revealed only in the conscious merger of these two polarities? Actually you can define computation, even universal machine, by using only addition and multiplication. So universal machine exists in elementary arithmetic in the same sense as in the existence of prime number. All the Bp and Dp are pure arithmetical sentences. What cannot be defined is Bp p, and we need to go out of the mind of the machine, and out of arithmetic, to provide the meaning, and machines can do that too. So, in arithmetic, you can find true statement about machine going outside of arithmetic. It is here that we have to be careful of not doing Searle's error of confusing levels, and that's why the epistemology internal in arithmetic can be bigger than arithmetic. Arithmetic itself does not believe in that epistemology, but it believes in numbers believing in them. Whatever you believe in will not been automatically believed by God, but God will always believe that you do believe in them. Bruno David Hi Bruno, You seem to not understand the role that the physical plays at all! This reminds me of an inversion of how most people cannot understand the way that math is abstract and have to work very hard to understand notions like in principle a coffee cup is the same as a doughnut. On 1/14/2012 6:58 AM, Bruno Marchal wrote: On 13 Jan 2012, at 18:24, Stephen P. King wrote: Hi Bruno, On 1/13/2012 4:38 AM, Bruno Marchal wrote: Hi Stephen, On 13 Jan 2012, at 00:58, Stephen P. King wrote: Hi Bruno, On 1/12/2012 1:01 PM, Bruno Marchal wrote: On 11 Jan 2012, at 19:35, acw wrote: On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. You can ascribe a sort of local consciousness to the person living, relatively to you, that Sigma_1 truth, but the person itself is really related to all the proofs (in Platonia) of that sentences (roughly speaking). OK, but that requires that I have a justification for a belief in Platonia. The closest that I can get to Platonia is something like the class of all verified proofs (which supervenes on some form of physical process.) You need just to believe that in
Re: Question about PA and 1p
On 16 January 2012 10:04, Bruno Marchal marc...@ulb.ac.be wrote: Actually you can define computation, even universal machine, by using only addition and multiplication. So universal machine exists in elementary arithmetic in the same sense as in the existence of prime number. That may be, but we were discussing interpretation. As you say above: YOU can define computation, even universal machine, by using only addition and multiplication (my emphasis). But this is surely, as you are wont to say, too quick. Firstly, in what sense can numbers in simple arithmetical relation define THEMSELVES as computation, or indeed as anything else than what they simply are? I think that the ascription of self-interpretation to a bare ontology is superficial; it conceals an implicit supplementary appeal to epistemology, and indeed to a self. Hence it appears that some perspectival union of epistemology and ontology is a prerequisite of interpretation. David On 14 Jan 2012, at 18:51, David Nyman wrote: On 14 January 2012 16:50, Stephen P. King stephe...@charter.net wrote: The problem is that mathematics cannot represent matter other than by invariance with respect to time, etc. absent an interpreter. Sure, but do you mean to say that the interpreter must be physical? I don't see why. And yet, as you say, the need for interpretation is unavoidable. Now, my understanding of Bruno, after some fairly close questioning (which may still leave me confused, of course) is that the elements of his arithmetical ontology are strictly limited to numbers (or their equivalent) + addition and multiplication. This emerged during discussion of macroscopic compositional principles implicit in the interpretation of micro-physical schemas; principles which are rarely understood as being epistemological in nature. Hence, strictly speaking, even the ascription of the notion of computation to arrangements of these bare arithmetical elements assumes further compositional principles and therefore appeals to some supplementary epistemological interpretation. In other words, any bare ontological schema, uninterpreted, is unable, from its own unsupplemented resources, to actualise whatever higher-level emergents may be implicit within it. But what else could deliver that interpretation/actualisation? What could embody the collapse of ontology and epistemology into a single actuality? Could it be that interpretation is finally revealed only in the conscious merger of these two polarities? Actually you can define computation, even universal machine, by using only addition and multiplication. So universal machine exists in elementary arithmetic in the same sense as in the existence of prime number. All the Bp and Dp are pure arithmetical sentences. What cannot be defined is Bp p, and we need to go out of the mind of the machine, and out of arithmetic, to provide the meaning, and machines can do that too. So, in arithmetic, you can find true statement about machine going outside of arithmetic. It is here that we have to be careful of not doing Searle's error of confusing levels, and that's why the epistemology internal in arithmetic can be bigger than arithmetic. Arithmetic itself does not believe in that epistemology, but it believes in numbers believing in them. Whatever you believe in will not been automatically believed by God, but God will always believe that you do believe in them. Bruno David Hi Bruno, You seem to not understand the role that the physical plays at all! This reminds me of an inversion of how most people cannot understand the way that math is abstract and have to work very hard to understand notions like in principle a coffee cup is the same as a doughnut. On 1/14/2012 6:58 AM, Bruno Marchal wrote: On 13 Jan 2012, at 18:24, Stephen P. King wrote: Hi Bruno, On 1/13/2012 4:38 AM, Bruno Marchal wrote: Hi Stephen, On 13 Jan 2012, at 00:58, Stephen P. King wrote: Hi Bruno, On 1/12/2012 1:01 PM, Bruno Marchal wrote: On 11 Jan 2012, at 19:35, acw wrote: On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something
Re: Question about PA and 1p
On 16 Jan 2012, at 15:32, David Nyman wrote: On 16 January 2012 10:04, Bruno Marchal marc...@ulb.ac.be wrote: Actually you can define computation, even universal machine, by using only addition and multiplication. So universal machine exists in elementary arithmetic in the same sense as in the existence of prime number. That may be, but we were discussing interpretation. As you say above: YOU can define computation, even universal machine, by using only addition and multiplication (my emphasis). Not just ME. A tiny part of arithmetic can too. All universal numbers can do that. No need of first person notion. All this can be shown in a 3p way. Indeed, in arithmetic. Even without the induction axioms, so that we don't need Löbian machine. The existence of the UD for example, is a theorem of (Robinson) arithmetic. Now, that kinds of truth are rather long and tedious to show. This was shown mainly by Gödel in his 1931 paper (for rich Löbian theories). It is called arithmetization of meta-mathematics. I will try to explain the salt of it without being too much technical below. But this is surely, as you are wont to say, too quick. Firstly, in what sense can numbers in simple arithmetical relation define THEMSELVES as computation, or indeed as anything else than what they simply are? Here you ask a more difficult question. Nevertheless it admits a positive answer. I think that the ascription of self-interpretation to a bare ontology is superficial; it conceals an implicit supplementary appeal to epistemology, and indeed to a self. But can define a notion of 3-self in arithmetic. Then to get the 1- self, we go at the meta-level and combine it with the notion of arithmetical truth. That notion is NOT definable in arithmetic, but that is a good thing, because it will explain why the notion of first person, and of consciousness, will not be definable by machine. Hence it appears that some perspectival union of epistemology and ontology is a prerequisite of interpretation. OK. But the whole force of comp comes from the fact that you can define a big part of that epistemology using only the elementary ontology. Let us agree on what we mean by defining something in arithmetic (or in the arithmetical language). The arithmetical language is the first order (predicate) logic with equality(=), so that it has the usual logical connectives (, V, -, ~ (and, or, implies, not), and the quantifiers E and A, (it exists and for all), together with the special arithmetical symbols 0, s + and *. To illustrate an arithmetical definition, let me give you some definitions of simple concepts. We can define the arithmetical relation x = y (x is less than or equal to y). Indeed x = y if and only if Ez(x+z = y) We can define x y (x is strictly less than y) by Ez((x+z) + s(0) = y) We can define (x divide y) by Ez(x*z = y) Now we can define (x is a prime number) by Az[ (x ≠ 1) and ((z divide x) - ((z = 1) or (z = x))] Which should be seen as a macro abbreviation of Az(~(x = s(0)) ((Ey(x*y = x) - (z = 1) V (z = x)). Now I tell you that we can define, exactly in that manner, the notion of universal number, computations, proofs, etc. In particular any proposition of the form phi_i(j) = k can be translated in arithmetic. A famous predicate due to Kleene is used for that effect . A universal number u can be defined by the relation AxAy(phi_u(x,y) = phi_x(y)), with x,y being a computable bijection from NXN to N. Like metamathematics can be arithmetized, theoretical computer science can be arithmetized. The interpretation is not done by me, but by the true relation between the numbers. 4 6 because it is true that Ez(s(s(s(s(0+z + s(0) = s(s(s(s(s(s(0)) ). That is true. Such a z exists, notably z = s(0). Likewize, assuming comp, the reason why you are conscious here and now is that your relative computational state exists, together with the infinitely many computations going through it. Your consciousness is harder to tackle, because it will refer more explicitly on that truth, like in the Bp p Theatetical trick. I do not need an extra God or observer of arithmetical truth, to interpret some number relation as computations, because the numbers, relatively to each other, already do that task. From their view, to believe that we need some extra-interpreter, would be like to believe that if your own brain is not observed by someone, it would not be conscious. Let me say two or three words on the SELF. Basically, it is very simple. You don't need universal numbers, nor super rich environment. You need an environment (machine, number) capable of duplicating, or concatenating piece of code. I usually sing this: If D(x) gives the description of x(x), then D(D) gives the description of DD. This belongs to the diagonalization family, and can be used to proves the existence of programs
Re: Question about PA and 1p
On 16 January 2012 18:08, Bruno Marchal marc...@ulb.ac.be wrote: I do not need an extra God or observer of arithmetical truth, to interpret some number relation as computations, because the numbers, relatively to each other, already do that task. From their view, to believe that we need some extra-interpreter, would be like to believe that if your own brain is not observed by someone, it would not be conscious. I'm unclear from the above - and indeed from the rest of your comments - whether you are defining interpretation in a purely 3p way, or whether you are implicitly placing it in a 1-p framework - e.g. where you say above From their view. If you do indeed assume that numbers can have such views, then I see why you would say that they interpret themselves, because adopting the 1p view is already to invoke a kind of emergence of number-epistemology. But such an emergence is still only a manner of speaking from OUR point of view, in that I can rephrase what you say above thus: From their view, to believe that THEY need some extra-interpreter... without taking such a point of view in any literal sense. Are you saying that consciousness somehow elevates number-epistemology into strong emergence, such that their point of view and self-interpretation become indistinguishable from my own? David On 16 Jan 2012, at 15:32, David Nyman wrote: On 16 January 2012 10:04, Bruno Marchal marc...@ulb.ac.be wrote: Actually you can define computation, even universal machine, by using only addition and multiplication. So universal machine exists in elementary arithmetic in the same sense as in the existence of prime number. That may be, but we were discussing interpretation. As you say above: YOU can define computation, even universal machine, by using only addition and multiplication (my emphasis). Not just ME. A tiny part of arithmetic can too. All universal numbers can do that. No need of first person notion. All this can be shown in a 3p way. Indeed, in arithmetic. Even without the induction axioms, so that we don't need Löbian machine. The existence of the UD for example, is a theorem of (Robinson) arithmetic. Now, that kinds of truth are rather long and tedious to show. This was shown mainly by Gödel in his 1931 paper (for rich Löbian theories). It is called arithmetization of meta-mathematics. I will try to explain the salt of it without being too much technical below. But this is surely, as you are wont to say, too quick. Firstly, in what sense can numbers in simple arithmetical relation define THEMSELVES as computation, or indeed as anything else than what they simply are? Here you ask a more difficult question. Nevertheless it admits a positive answer. I think that the ascription of self-interpretation to a bare ontology is superficial; it conceals an implicit supplementary appeal to epistemology, and indeed to a self. But can define a notion of 3-self in arithmetic. Then to get the 1-self, we go at the meta-level and combine it with the notion of arithmetical truth. That notion is NOT definable in arithmetic, but that is a good thing, because it will explain why the notion of first person, and of consciousness, will not be definable by machine. Hence it appears that some perspectival union of epistemology and ontology is a prerequisite of interpretation. OK. But the whole force of comp comes from the fact that you can define a big part of that epistemology using only the elementary ontology. Let us agree on what we mean by defining something in arithmetic (or in the arithmetical language). The arithmetical language is the first order (predicate) logic with equality(=), so that it has the usual logical connectives (, V, -, ~ (and, or, implies, not), and the quantifiers E and A, (it exists and for all), together with the special arithmetical symbols 0, s + and *. To illustrate an arithmetical definition, let me give you some definitions of simple concepts. We can define the arithmetical relation x = y (x is less than or equal to y). Indeed x = y if and only if Ez(x+z = y) We can define x y (x is strictly less than y) by Ez((x+z) + s(0) = y) We can define (x divide y) by Ez(x*z = y) Now we can define (x is a prime number) by Az[ (x ≠ 1) and ((z divide x) - ((z = 1) or (z = x))] Which should be seen as a macro abbreviation of Az(~(x = s(0)) ((Ey(x*y = x) - (z = 1) V (z = x)). Now I tell you that we can define, exactly in that manner, the notion of universal number, computations, proofs, etc. In particular any proposition of the form phi_i(j) = k can be translated in arithmetic. A famous predicate due to Kleene is used for that effect . A universal number u can be defined by the relation AxAy(phi_u(x,y) = phi_x(y)), with x,y being a computable bijection from NXN to N. Like metamathematics can be arithmetized, theoretical computer science can be arithmetized. The interpretation is not done by
Re: Question about PA and 1p
Hi Bruno, You seem to not understand the role that the physical plays at all! This reminds me of an inversion of how most people cannot understand the way that math is abstract and have to work very hard to understand notions like in principle a coffee cup is the same as a doughnut. On 1/14/2012 6:58 AM, Bruno Marchal wrote: On 13 Jan 2012, at 18:24, Stephen P. King wrote: Hi Bruno, On 1/13/2012 4:38 AM, Bruno Marchal wrote: Hi Stephen, On 13 Jan 2012, at 00:58, Stephen P. King wrote: Hi Bruno, On 1/12/2012 1:01 PM, Bruno Marchal wrote: On 11 Jan 2012, at 19:35, acw wrote: On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. You can ascribe a sort of local consciousness to the person living, relatively to you, that Sigma_1 truth, but the person itself is really related to all the proofs (in Platonia) of that sentences (roughly speaking). OK, but that requires that I have a justification for a belief in Platonia. The closest that I can get to Platonia is something like the class of all verified proofs (which supervenes on some form of physical process.) You need just to believe that in the standard model of PA a sentence is true or false. I have not yet seen any book in math mentioning anything physical to define what that means. *All* math papers you cited assume no less. I cannot understand how such an obvious concept is not understood, even the notion of universality assumes it. The point is that mathematical statements require some form of physicality to be known and communicated, OK. But they does not need phyicality to be just true. That's the point. Surely, but the truthfulness of a mathematical statement is meaningless without the possibility of physical implementation. One cannot even know of it absent the possibility of the physical. it just is the case that the sentence, model, recursive algorithm, whatever concept, etc. is independent of any particular form of physical implementation but is not independent of all physical representations. Of course it is. When you reason in PA you don't use any axiom referring to physics. To say that you need a physical brain begs the question *and* is a level-of-reasoning error. PA does need to have any axioms that refer to physics. The fact that PA is inferred from patterns of chalk on a chalk board or patterns of ink on a whiteboard or patterns of pixels on a computer monitor or patterns of scratches in the dust or ... is sufficient to establish the truth of what I am saying. If you remove the possibility of physical implementation you also remove the possibility of meaningfulness. We cannot completely abstract away the role played by the physical world. That's what we do in math. Yes, but all the while the physical world is the substrate for our patterns without which there is meaninglessness. I simply cannot see how Sigma_1 sentences can interface with each other such that one can know anything about another absent some form of physicality. The interfaces and the relative implementations are defined using addition and multiplication only, like in Gödel's original paper. Then UDA shows why physicality is an emergent pattern in the mind of number, and why it has to be like that if comp is true. AUDA shows how to make the derivation. No, you have only proven that the idea that the physicalist idea that mind is an epiphenomena is false, No. I show that the physical reality is not an ontological reality, once we assume we are (even material) machine. And I agree, the physical is not a primitive in the existential sense, but neither
Re: Question about PA and 1p
On 14 January 2012 16:50, Stephen P. King stephe...@charter.net wrote: The problem is that mathematics cannot represent matter other than by invariance with respect to time, etc. absent an interpreter. Sure, but do you mean to say that the interpreter must be physical? I don't see why. And yet, as you say, the need for interpretation is unavoidable. Now, my understanding of Bruno, after some fairly close questioning (which may still leave me confused, of course) is that the elements of his arithmetical ontology are strictly limited to numbers (or their equivalent) + addition and multiplication. This emerged during discussion of macroscopic compositional principles implicit in the interpretation of micro-physical schemas; principles which are rarely understood as being epistemological in nature. Hence, strictly speaking, even the ascription of the notion of computation to arrangements of these bare arithmetical elements assumes further compositional principles and therefore appeals to some supplementary epistemological interpretation. In other words, any bare ontological schema, uninterpreted, is unable, from its own unsupplemented resources, to actualise whatever higher-level emergents may be implicit within it. But what else could deliver that interpretation/actualisation? What could embody the collapse of ontology and epistemology into a single actuality? Could it be that interpretation is finally revealed only in the conscious merger of these two polarities? David Hi Bruno, You seem to not understand the role that the physical plays at all! This reminds me of an inversion of how most people cannot understand the way that math is abstract and have to work very hard to understand notions like in principle a coffee cup is the same as a doughnut. On 1/14/2012 6:58 AM, Bruno Marchal wrote: On 13 Jan 2012, at 18:24, Stephen P. King wrote: Hi Bruno, On 1/13/2012 4:38 AM, Bruno Marchal wrote: Hi Stephen, On 13 Jan 2012, at 00:58, Stephen P. King wrote: Hi Bruno, On 1/12/2012 1:01 PM, Bruno Marchal wrote: On 11 Jan 2012, at 19:35, acw wrote: On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. You can ascribe a sort of local consciousness to the person living, relatively to you, that Sigma_1 truth, but the person itself is really related to all the proofs (in Platonia) of that sentences (roughly speaking). OK, but that requires that I have a justification for a belief in Platonia. The closest that I can get to Platonia is something like the class of all verified proofs (which supervenes on some form of physical process.) You need just to believe that in the standard model of PA a sentence is true or false. I have not yet seen any book in math mentioning anything physical to define what that means. *All* math papers you cited assume no less. I cannot understand how such an obvious concept is not understood, even the notion of universality assumes it. The point is that mathematical statements require some form of physicality to be known and communicated, OK. But they does not need phyicality to be just true. That's the point. Surely, but the truthfulness of a mathematical statement is meaningless without the possibility of physical implementation. One cannot even know of it absent the possibility of the physical. it just is the case that the sentence, model, recursive algorithm, whatever concept, etc. is independent of any particular form of physical implementation but is not independent of all physical representations. Of course it is. When you reason in PA you don't use any axiom referring to physics. To say that you
Re: Question about PA and 1p
On 1/14/2012 10:41 AM, Stephen P. King wrote: I suppose that that is the case, but how do mathematical entities implement themselves other than via physical processes? We seem to be thinking that this is a solvable Chicken and Egg problem and I argue that we cannot use the argument of reduction to solve it. We must have both the physical and the mental, not at the primitive level of existence to be sure, but at the level where they have meaning. Suppose there are characters in a computer game that have very sophisticated AI. Don't events in the game have meaning for them? The meaning is implicit in the actions and reactions. Brent This is why I argue for a form of dualism that transforms into a neutral monism, like that of Russel, when taken to the level of ding and sich. At teh level of ding and sich difference itself vanishes and thus to argue that matter or number is primitive is a mute point. We must be careful that we are not collapsing the levels in our thinking about this. -- 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: Question about PA and 1p
On 1/14/2012 4:05 PM, meekerdb wrote: On 1/14/2012 10:41 AM, Stephen P. King wrote: I suppose that that is the case, but how do mathematical entities implement themselves other than via physical processes? We seem to be thinking that this is a solvable Chicken and Egg problem and I argue that we cannot use the argument of reduction to solve it. We must have both the physical and the mental, not at the primitive level of existence to be sure, but at the level where they have meaning. Suppose there are characters in a computer game that have very sophisticated AI. Don't events in the game have meaning for them? The meaning is implicit in the actions and reactions. Brent Hi Brent, Let us consider your idea carefully as you are asking an important question, I think. Those NPC (non-player characters), is their behavior the result of a finite list of if X then Y statements or equivalents? Where does the possibility of to whom-ness lie for that list of if then statements? How does a per-specified list of properties encode a sense of self? Forget the anthropomorphic stuff, lets focus on the 1p stuff here. How do we bridge between the per-specified list of if then's to a coherent notion of 1p? Onward! Stephen -- 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: Question about PA and 1p
On 1/14/2012 6:21 PM, Stephen P. King wrote: On 1/14/2012 4:05 PM, meekerdb wrote: On 1/14/2012 10:41 AM, Stephen P. King wrote: I suppose that that is the case, but how do mathematical entities implement themselves other than via physical processes? We seem to be thinking that this is a solvable Chicken and Egg problem and I argue that we cannot use the argument of reduction to solve it. We must have both the physical and the mental, not at the primitive level of existence to be sure, but at the level where they have meaning. Suppose there are characters in a computer game that have very sophisticated AI. Don't events in the game have meaning for them? The meaning is implicit in the actions and reactions. Brent Hi Brent, Let us consider your idea carefully as you are asking an important question, I think. Those NPC (non-player characters), is their behavior the result of a finite list of if X then Y statements or equivalents? Dunno. If I were writing it I'd probably throw in a little randomness as well as functions with self-modification to allow learning. Where does the possibility of to whom-ness lie for that list of if then statements? I don't know what to whom-ness means. How does a per-specified list of properties encode a sense of self? I'm not sure what you mean by sense of self. The AI would encode the position and state of the character, including values, plans, self evaluation, etc. Forget the anthropomorphic stuff, lets focus on the 1p stuff here. How do we bridge between the per-specified list of if then's to a coherent notion of 1p? By making the AI behave like a person. How do you know there's a gap to be bridged? Brent -- 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: Question about PA and 1p
On 1/15/2012 1:07 AM, meekerdb wrote: On 1/14/2012 6:21 PM, Stephen P. King wrote: On 1/14/2012 4:05 PM, meekerdb wrote: On 1/14/2012 10:41 AM, Stephen P. King wrote: I suppose that that is the case, but how do mathematical entities implement themselves other than via physical processes? We seem to be thinking that this is a solvable Chicken and Egg problem and I argue that we cannot use the argument of reduction to solve it. We must have both the physical and the mental, not at the primitive level of existence to be sure, but at the level where they have meaning. Suppose there are characters in a computer game that have very sophisticated AI. Don't events in the game have meaning for them? The meaning is implicit in the actions and reactions. Brent Hi Brent, Let us consider your idea carefully as you are asking an important question, I think. Those NPC (non-player characters), is their behavior the result of a finite list of if X then Y statements or equivalents? Dunno. If I were writing it I'd probably throw in a little randomness as well as functions with self-modification to allow learning. How would these not included in the finite list of If - then rules? Where does the possibility of to whom-ness lie for that list of if then statements? I don't know what to whom-ness means. Speculate what I might mean... How does a per-specified list of properties encode a sense of self? I'm not sure what you mean by sense of self. The AI would encode the position and state of the character, including values, plans, self evaluation, etc. How do you encode a map of L has M properties including location in a way that is updatable, or equivalently, for the fixed (with respect to virtual location) how do you encode changes in the environment with respect to the system such that there is a finite upper bound on the recursions of maps within maps? It seems to me that a sense of self is at least some form of model that quantifies the distinctions of what it is versus what it is not. Some set membership function would work, maybe. But this too seems to be encodable in if then rules... Forget the anthropomorphic stuff, lets focus on the 1p stuff here. How do we bridge between the per-specified list of if then's to a coherent notion of 1p? By making the AI behave like a person. How do you know there's a gap to be bridged? What is a person? Beware of circular definitions! I am not assuming a gap, I am just trying to reason through this thought experiment with you. Onward! Stephen Brent -- 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: Question about PA and 1p
On 1/14/2012 10:32 PM, Stephen P. King wrote: On 1/15/2012 1:07 AM, meekerdb wrote: On 1/14/2012 6:21 PM, Stephen P. King wrote: On 1/14/2012 4:05 PM, meekerdb wrote: On 1/14/2012 10:41 AM, Stephen P. King wrote: I suppose that that is the case, but how do mathematical entities implement themselves other than via physical processes? We seem to be thinking that this is a solvable Chicken and Egg problem and I argue that we cannot use the argument of reduction to solve it. We must have both the physical and the mental, not at the primitive level of existence to be sure, but at the level where they have meaning. Suppose there are characters in a computer game that have very sophisticated AI. Don't events in the game have meaning for them? The meaning is implicit in the actions and reactions. Brent Hi Brent, Let us consider your idea carefully as you are asking an important question, I think. Those NPC (non-player characters), is their behavior the result of a finite list of if X then Y statements or equivalents? Dunno. If I were writing it I'd probably throw in a little randomness as well as functions with self-modification to allow learning. How would these not included in the finite list of If - then rules? Where does the possibility of to whom-ness lie for that list of if then statements? I don't know what to whom-ness means. Speculate what I might mean... Why speculate when I can ask you? How does a per-specified list of properties encode a sense of self? I'm not sure what you mean by sense of self. The AI would encode the position and state of the character, including values, plans, self evaluation, etc. How do you encode a map of L has M properties including location in a way that is updatable, or equivalently, for the fixed (with respect to virtual location) how do you encode changes in the environment with respect to the system such that there is a finite upper bound on the recursions of maps within maps? I'm not sure what you're talking about? What's a map of L? Are you asking how to write an AI program? Whether to use hash tables or matrices or linked lists? However I encode changes in the environment, making the recursions finite is pretty much taken care of by hardware space and time limits. It seems to me that a sense of self is at least some form of model that quantifies the distinctions of what it is versus what it is not. Some set membership function would work, maybe. But this too seems to be encodable in if then rules... Forget the anthropomorphic stuff, lets focus on the 1p stuff here. How do we bridge between the per-specified list of if then's to a coherent notion of 1p? By making the AI behave like a person. How do you know there's a gap to be bridged? What is a person? Beware of circular definitions! I am and I guess you are. Brent I am not assuming a gap, I am just trying to reason through this thought experiment with you. Onward! Stephen Brent -- 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: Question about PA and 1p
Hi Stephen, On 13 Jan 2012, at 00:58, Stephen P. King wrote: Hi Bruno, On 1/12/2012 1:01 PM, Bruno Marchal wrote: On 11 Jan 2012, at 19:35, acw wrote: On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. You can ascribe a sort of local consciousness to the person living, relatively to you, that Sigma_1 truth, but the person itself is really related to all the proofs (in Platonia) of that sentences (roughly speaking). OK, but that requires that I have a justification for a belief in Platonia. The closest that I can get to Platonia is something like the class of all verified proofs (which supervenes on some form of physical process.) You need just to believe that in the standard model of PA a sentence is true or false. I have not yet seen any book in math mentioning anything physical to define what that means. *All* math papers you cited assume no less. I simply cannot see how Sigma_1 sentences can interface with each other such that one can know anything about another absent some form of physicality. The interfaces and the relative implementations are defined using addition and multiplication only, like in Gödel's original paper. Then UDA shows why physicality is an emergent pattern in the mind of number, and why it has to be like that if comp is true. AUDA shows how to make the derivation. If I take away all forms of physical means of communicating ideas, no chalkboards, paper, computer screens, etc., how can ideas be possibly communicated? Because arithmetical truth contains all machine 'dreams, including dreams of chalkboards, papers, screens, etc. UDA has shown that a real paper, or real screen is an emergent stable pattern supervening on infinities of computation, through a competition between all universal numbers occurring below our substitution level. You might try to tell me where in the proof you lost the arguement. Mere existence does not specify properties. That's not correct. We can explain the property being prime from the mere existence of 0, s(0), s(s(0)), ... and the recursive laws of addition and multiplication. I go so far as considering that the wavefunction and its unitary evolution exists and it is a sufficiently universal physical process to implement the UD, but the UD as just the equivalent to Integers, nay, that I cannot believe in. “One cannot speak about whatever one cannot talk.” ~ Maturana (1978, p. 49) I think Maturana was alluding to Wittgenstein, and that sentence is almost as ridiculous as Damascius saying one sentence about the ineffable is one sentence too much. But it is a deep meta-truth playing some role in number's theology. But I think that you cannot define the universal wave without postulating arithmetical realism. In fact real number+trigonometrical function is a stronger form of realism than arithmetical realism. Adding physical in front of it adds nothing but a magical notion of primary substance. Epistemologically it is a form of treachery, by UDA, it singles out a universal number and postulate it is real, when comp explains precisely that such a move cannot work. Bruno Bruno I think you might be confusing structures/relations which can be contained within PA with PA itself. On 1/11/2012 12:07 PM, acw wrote: On 1/10/2012 17:48, Bruno Marchal wrote: On 10 Jan 2012, at 12:58, acw wrote: On 1/10/2012 12:03, Bruno Marchal wrote: On 09 Jan 2012, at 19:36, acw wrote: To put it more simply: if Church Turing Thesis(CTT) is correct, mathematics is the same for any system or being you can imagine. I am not
Re: Question about PA and 1p
On 13 January 2012 17:24, Stephen P. King stephe...@charter.net wrote: I submit to you that you cannot just ignore the universals vs. nominal problem and posit by fiat that just because one can proof the truth of some statement that that statement's existence determines its properties. Our ability to communicate ideas follows from their universality, that they do not require *some particular* physical implementation, but that is not the same as requiring *no* physical implementation. You argue that *no* physical implementation is necessary; I disagree. Forgive me for butting in (particularly in the light of the fact that I too lack Bruno's erudition, only in spades) but I simply don't read Bruno's work in the way you are representing it. I see it like this: we have little option but to split our theories of what there is into two parts: the epistemological (i.e. the only form in which, and the exclusive means whereby, we have any access to information) and the ontological (i.e. some coherent theoretical framework in which to situate what that knowledge seems to reveal, and also, ideally, one that is able to account satisfactorily for how we are able to come by such knowledge in the first place). But after Kant, we can surely no longer believe that the ontological component of this dyad can possibly give us direct access to some ultimate ding and sich? Rather, what we seek in such theories is a mathematical schema in terms of which the relations between primitive theoretical entities, which themselves explicitly lack any further internal relations or characteristics, can be framed. Of course, this bare mathematical depiction cannot be reconciled with any aspect of experience without recombination with the epistemological component, which in most theories typically entails a sleight-of-thought that is still, to say the least, almost entirely opaque. If the foregoing is even vaguely true, then surely your debate with Bruno cannot be about whether either matter or numbers really exist, because the very notion of real existence transcends anything about which we can theorise or have experience. Since mathematics delimits any possible ontological characterisation, the debate can in consequence only be about the derivation, priority and hence relative primitiveness, of the mathematical entities thus characterised. In fact, this is an implicit assumption, so far as I can see, amongst physicists, who have until quite recently assumed that the mathematical structure of physics, as currently known, simply was the relevant primitive structure. However, attempts to reach beyond the puzzles of current theory have already led some, like Tegmark, to an explicitly mathematical characterisation of physical ontology. Bruno's work, it seems to me, is in the same spirit, with the critical distinction that he believes that, unless the epistemological component is placed at the centre of the theory, the appearances cannot ultimately be saved. Consequently, it is inaccurate to say that physical representation is not a core aspect of his theory - it is absolutely central, just not primitive, in the sense that the theory seeks to derive it as an aspect of a more fundamental (in fact, in Bruno's contention, the MOST fundamental) mathematical framework . David Hi Bruno, On 1/13/2012 4:38 AM, Bruno Marchal wrote: Hi Stephen, On 13 Jan 2012, at 00:58, Stephen P. King wrote: Hi Bruno, On 1/12/2012 1:01 PM, Bruno Marchal wrote: On 11 Jan 2012, at 19:35, acw wrote: On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. You can ascribe a sort of local consciousness to the person living, relatively
Re: Question about PA and 1p
On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. I think you might be confusing structures/relations which can be contained within PA with PA itself. On 1/11/2012 12:07 PM, acw wrote: On 1/10/2012 17:48, Bruno Marchal wrote: On 10 Jan 2012, at 12:58, acw wrote: On 1/10/2012 12:03, Bruno Marchal wrote: On 09 Jan 2012, at 19:36, acw wrote: To put it more simply: if Church Turing Thesis(CTT) is correct, mathematics is the same for any system or being you can imagine. I am not sure why. Sigma_1 arithmetic would be the same; but higher mathematics (set theory, analysis) might still be different. If it's wrong, maybe stuff like concrete infinities, hypercomputation and infinite minds could exist and that would falsify COMP, however there is zero evidence for any of that being possible. Not sure, if CT is wrong, there would be finite machines, working in finite time, with well defined instructions, which would be NOT Turing emulable. Hypercomputation and infinite (human) minds would contradict comp, not CT. On the contrary, they need CT to claim that they compute more than any programmable machines. CT is part of comp, but comp is not part of CT. Beyond this, I agree with your reply to Craig. In that response I was using CT in the more unrestricted form: all effectively computable functions are Turing-computable. I understand, but that is confusing. David Deutsch and many physicists are a bit responsible of that confusion, by attempting to have a notion of effectivity relying on physics. The original statement of Church, Turing, Markov, Post, ... concerns only the intuitively human computable functions, or the functions computable by finitary means. It asserts that the class of such intuitively computable functions is the same as the class of functions computable by some Turing machine (or by the unique universal Turing machine). Such a notion is a priori completely independent of the notion of computable by physical means. Yes, with the usual notion of Turing-computable, you don't really need more than arithmetic. It might be a bit stronger than the usual equivalency proofs between a very wide range of models of computation (Turing machines, Abacus/PA machines, (primitive) recursive functions (+minimization), all kinds of more current models of computation, languages and so on). Yes. I even suspect that CT makes the class of functions computable by physics greater than the class of Church. That could be possible, but more evidence is needed for this(beyond the random oracle). I also wonder 2 other things: 1) would we be able to really know if we find ourselves in such a world (I'm leaning toward unlikely, but I'm agnostic about this) 2) would someone performing my experiment(described in another message), lose the ability to find himself in such a world (I'm leaning toward 'no, if it's possible now, it should still be possible'). If hypercomputation was actually possible that would mean that strong variant of CT would be false, because there would be something effectively computable that wasn't computable by a Turing machine. OK. In a way, that strong form of CT might already be false with comp, only in the 1p sense as you get a free random oracle as well as always staying consistent(and 'alive'), but it's not false in the 3p view... Yes. Comp makes physics a first person plural reality, and a priori we might be able to exploit the first plural indeterminacy to compute more function, like we know already that we have more processes, like that free random oracle. The empirical fact that quantum computer does not violate CT can make us doubt about this. In the third
Re: Question about PA and 1p
Hi David, I do appreciate your remarks and thank you for writing them up and posting them. Let me interleave some comments in reply. On 1/13/2012 1:43 PM, David Nyman wrote: On 13 January 2012 17:24, Stephen P. Kingstephe...@charter.net wrote: I submit to you that you cannot just ignore the universals vs. nominal problem and posit by fiat that just because one can proof the truth of some statement that that statement's existence determines its properties. Our ability to communicate ideas follows from their universality, that they do not require *some particular* physical implementation, but that is not the same as requiring *no* physical implementation. You argue that *no* physical implementation is necessary; I disagree. Forgive me for butting in (particularly in the light of the fact that I too lack Bruno's erudition, only in spades) but I simply don't read Bruno's work in the way you are representing it. I see it like this: we have little option but to split our theories of what there is into two parts: the epistemological (i.e. the only form in which, and the exclusive means whereby, we have any access to information) and the ontological (i.e. some coherent theoretical framework in which to situate what that knowledge seems to reveal, and also, ideally, one that is able to account satisfactorily for how we are able to come by such knowledge in the first place). My point is that our epistemological and ontological theories are predicated upon our actuality (not just existence) as physical systems that have the ability to reason. It is obviously true that if something that is like an observer does not exist then none of this discussion would exist either. We simply cannot remove ourselves from our theories, concepts, models, representations, ... I am trying to point out that the same holds for physical implementations of those theories, concepts, models, representations, ... Consider how the notion of meaningfulness implicitly requires at to whom a meaning obtains. But there is more to this discussion But after Kant, we can surely no longer believe that the ontological component of this dyad can possibly give us direct access to some ultimate ding and sich? Right, we can show via a logical argument that we cannot have knowledge of any ding and sich by any direct means, I will not go into such for sake of brevity, but we need some way to get around this fact. We postulate assumptions when we are theory making and see where they take us... Rather, what we seek in such theories is a mathematical schema in terms of which the relations between primitive theoretical entities, which themselves explicitly lack any further internal relations or characteristics, can be framed. OK, but this remark itself assumes an ontological postulate! What about models that do not assume primitive theoretical entities, which themselves explicitly lack any further internal relations or characteristics..? There are theories, such as what Jon Barwise et al discussed in his papers and books, that do not assume the well-founded axiom http://en.wikipedia.org/wiki/Well-foundedness (aka Axiom of regularity) or equivalent. Non-Well Founded set theory http://plato.stanford.edu/entries/nonwellfounded-set-theory/exists and works! If and when we base our ideas about Existence, Reality and the nature and means of knowledge on entities such as numbers, as Bruno is doing, then we are implicitly assuming a particular mereology http://plato.stanford.edu/entries/mereology/ (relationship between wholes and parts) when, given the existence of alternatives (given that we can mathematically prove their properties follow from blah blah blah..). My argument rest on the fact that other schemata are possible! That there are mathematical models that do not require a notion of a primitive (in the Greek sense of Atoms, as being indivisible and lacking of any internal relations or characteristics) but instead consider entities as, crudely explained, composed of others. This idea has been long castigated as implying all kinds of problems and paradox such as the Cretan Liar, Sets that both contain and do not contain themselves, etc. But I content that all of these pathologies follow from the failure of thinkers to comprehend the deep implications of what it means for a statement, claim, Sigma_1 sentence, etc. to have meaningfulness. There is always an implicit to whom meaning obtains and that to whom-ness cannot be separated from the ding and sich-ness of objects, be they planets, numbers, or Pink Polka-dotted Unicorns. Of course, this bare mathematical depiction cannot be reconciled with any aspect of experience without recombination with the epistemological component, which in most theories typically entails a sleight-of-thought that is still, to say the least, almost entirely opaque. I agree! This, I argue, is the underlying reason why I am making a big
Re: Question about PA and 1p
Hi Stephen Thanks for responding to my post in such detail. I'll need some time to digest your points, although I'm not at all sure I have the necessary background to grasp all of what you are saying. However, I would just like to remark at this point that my characterisation of the sought-for ontology as mathematical is not because I have any special insight into the matter (pun intended) - how could I? Rather it is because I observe that such an assumption seems to have become, either implicitly or explicitly, the principal way in which physics - the default ontology of modern science - is characterised. The determined objectivity of this approach may indeed obscure key problems at the heart of the interpretation of the resulting formalism, but it's all too easy to ignore or trivialise these when one is in the grip of a doctrine. As to Bruno's position, given that his point of departure is the computational theory of mind, he argues, if I understand him, that this consequently places particular logical constraints on his choice of ontology from the outset. Does this imply that you explicitly reject CTM, or do you rather disagree about the ontological constraints it might imply? Or, if your own theoretical point of entry begins from quite different basic assumptions, what would be the most straightforward introduction to these? David On 13 January 2012 21:26, Stephen P. King stephe...@charter.net wrote: Hi David, I do appreciate your remarks and thank you for writing them up and posting them. Let me interleave some comments in reply. On 1/13/2012 1:43 PM, David Nyman wrote: On 13 January 2012 17:24, Stephen P. King stephe...@charter.net wrote: I submit to you that you cannot just ignore the universals vs. nominal problem and posit by fiat that just because one can proof the truth of some statement that that statement's existence determines its properties. Our ability to communicate ideas follows from their universality, that they do not require *some particular* physical implementation, but that is not the same as requiring *no* physical implementation. You argue that *no* physical implementation is necessary; I disagree. Forgive me for butting in (particularly in the light of the fact that I too lack Bruno's erudition, only in spades) but I simply don't read Bruno's work in the way you are representing it. I see it like this: we have little option but to split our theories of what there is into two parts: the epistemological (i.e. the only form in which, and the exclusive means whereby, we have any access to information) and the ontological (i.e. some coherent theoretical framework in which to situate what that knowledge seems to reveal, and also, ideally, one that is able to account satisfactorily for how we are able to come by such knowledge in the first place). My point is that our epistemological and ontological theories are predicated upon our actuality (not just existence) as physical systems that have the ability to reason. It is obviously true that if something that is like an observer does not exist then none of this discussion would exist either. We simply cannot remove ourselves from our theories, concepts, models, representations, ... I am trying to point out that the same holds for physical implementations of those theories, concepts, models, representations, ... Consider how the notion of meaningfulness implicitly requires at to whom a meaning obtains. But there is more to this discussion But after Kant, we can surely no longer believe that the ontological component of this dyad can possibly give us direct access to some ultimate ding and sich? Right, we can show via a logical argument that we cannot have knowledge of any ding and sich by any direct means, I will not go into such for sake of brevity, but we need some way to get around this fact. We postulate assumptions when we are theory making and see where they take us... Rather, what we seek in such theories is a mathematical schema in terms of which the relations between primitive theoretical entities, which themselves explicitly lack any further internal relations or characteristics, can be framed. OK, but this remark itself assumes an ontological postulate! What about models that do not assume primitive theoretical entities, which themselves explicitly lack any further internal relations or characteristics..? There are theories, such as what Jon Barwise et al discussed in his papers and books, that do not assume the well-founded axiom (aka Axiom of regularity) or equivalent. Non-Well Founded set theory exists and works! If and when we base our ideas about Existence, Reality and the nature and means of knowledge on entities such as numbers, as Bruno is doing, then we are implicitly assuming a particular mereology (relationship between wholes and parts) when, given the existence of alternatives (given that we can mathematically
Re: Question about PA and 1p
On 1/13/2012 7:29 PM, David Nyman wrote: Hi Stephen Thanks for responding to my post in such detail. I'll need some time to digest your points, although I'm not at all sure I have the necessary background to grasp all of what you are saying. However, I would just like to remark at this point that my characterisation of the sought-for ontology as mathematical is not because I have any special insight into the matter (pun intended) - how could I? Rather it is because I observe that such an assumption seems to have become, either implicitly or explicitly, the principal way in which physics - the default ontology of modern science - is characterised. The determined objectivity of this approach may indeed obscure key problems at the heart of the interpretation of the resulting formalism, but it's all too easy to ignore or trivialise these when one is in the grip of a doctrine. As to Bruno's position, given that his point of departure is the computational theory of mind, he argues, if I understand him, that this consequently places particular logical constraints on his choice of ontology from the outset. Does this imply that you explicitly reject CTM, or do you rather disagree about the ontological constraints it might imply? Or, if your own theoretical point of entry begins from quite different basic assumptions, what would be the most straightforward introduction to these? David Hi David, I am coming from a very different point of view. I ask that you take a look at the non-well founded set theory stuff and see if you can figure out for yourself its implications re ontology. I very well might be reading something into it that is not there, but after having read almost all of Barwise et al's books (particularly The Liar: An Essay on Truth and Circularity http://www.amazon.com/Liar-Essay-Truth-Circularity/dp/0195059441/ref=ntt_at_ep_dpt_4 and Vicious Circles http://www.amazon.com/Vicious-Circles-Center-Language-Information/dp/1575860082/ref=ntt_at_ep_dpt_2 ) and studying metamathematics and philosophy (a wide swath), it seems to me that my conclusion is correct. OTOH, I know that I am missing something as I am fallible and finite. ;-) Onward! Stephen On 13 January 2012 21:26, Stephen P. Kingstephe...@charter.net wrote: Hi David, I do appreciate your remarks and thank you for writing them up and posting them. Let me interleave some comments in reply. On 1/13/2012 1:43 PM, David Nyman wrote: On 13 January 2012 17:24, Stephen P. Kingstephe...@charter.net wrote: I submit to you that you cannot just ignore the universals vs. nominal problem and posit by fiat that just because one can proof the truth of some statement that that statement's existence determines its properties. Our ability to communicate ideas follows from their universality, that they do not require *some particular* physical implementation, but that is not the same as requiring *no* physical implementation. You argue that *no* physical implementation is necessary; I disagree. Forgive me for butting in (particularly in the light of the fact that I too lack Bruno's erudition, only in spades) but I simply don't read Bruno's work in the way you are representing it. I see it like this: we have little option but to split our theories of what there is into two parts: the epistemological (i.e. the only form in which, and the exclusive means whereby, we have any access to information) and the ontological (i.e. some coherent theoretical framework in which to situate what that knowledge seems to reveal, and also, ideally, one that is able to account satisfactorily for how we are able to come by such knowledge in the first place). My point is that our epistemological and ontological theories are predicated upon our actuality (not just existence) as physical systems that have the ability to reason. It is obviously true that if something that is like an observer does not exist then none of this discussion would exist either. We simply cannot remove ourselves from our theories, concepts, models, representations, ... I am trying to point out that the same holds for physical implementations of those theories, concepts, models, representations, ... Consider how the notion of meaningfulness implicitly requires at to whom a meaning obtains. But there is more to this discussion But after Kant, we can surely no longer believe that the ontological component of this dyad can possibly give us direct access to some ultimate ding and sich? Right, we can show via a logical argument that we cannot have knowledge of any ding and sich by any direct means, I will not go into such for sake of brevity, but we need some way to get around this fact. We postulate assumptions when we are theory making and see where they take us... Rather, what we seek in such theories is a mathematical schema in terms of which the relations between primitive theoretical entities, which themselves explicitly lack any further
Re: Question about PA and 1p
On 11 Jan 2012, at 19:35, acw wrote: On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. You can ascribe a sort of local consciousness to the person living, relatively to you, that Sigma_1 truth, but the person itself is really related to all the proofs (in Platonia) of that sentences (roughly speaking). Bruno I think you might be confusing structures/relations which can be contained within PA with PA itself. On 1/11/2012 12:07 PM, acw wrote: On 1/10/2012 17:48, Bruno Marchal wrote: On 10 Jan 2012, at 12:58, acw wrote: On 1/10/2012 12:03, Bruno Marchal wrote: On 09 Jan 2012, at 19:36, acw wrote: To put it more simply: if Church Turing Thesis(CTT) is correct, mathematics is the same for any system or being you can imagine. I am not sure why. Sigma_1 arithmetic would be the same; but higher mathematics (set theory, analysis) might still be different. If it's wrong, maybe stuff like concrete infinities, hypercomputation and infinite minds could exist and that would falsify COMP, however there is zero evidence for any of that being possible. Not sure, if CT is wrong, there would be finite machines, working in finite time, with well defined instructions, which would be NOT Turing emulable. Hypercomputation and infinite (human) minds would contradict comp, not CT. On the contrary, they need CT to claim that they compute more than any programmable machines. CT is part of comp, but comp is not part of CT. Beyond this, I agree with your reply to Craig. In that response I was using CT in the more unrestricted form: all effectively computable functions are Turing-computable. I understand, but that is confusing. David Deutsch and many physicists are a bit responsible of that confusion, by attempting to have a notion of effectivity relying on physics. The original statement of Church, Turing, Markov, Post, ... concerns only the intuitively human computable functions, or the functions computable by finitary means. It asserts that the class of such intuitively computable functions is the same as the class of functions computable by some Turing machine (or by the unique universal Turing machine). Such a notion is a priori completely independent of the notion of computable by physical means. Yes, with the usual notion of Turing-computable, you don't really need more than arithmetic. It might be a bit stronger than the usual equivalency proofs between a very wide range of models of computation (Turing machines, Abacus/PA machines, (primitive) recursive functions (+minimization), all kinds of more current models of computation, languages and so on). Yes. I even suspect that CT makes the class of functions computable by physics greater than the class of Church. That could be possible, but more evidence is needed for this(beyond the random oracle). I also wonder 2 other things: 1) would we be able to really know if we find ourselves in such a world (I'm leaning toward unlikely, but I'm agnostic about this) 2) would someone performing my experiment(described in another message), lose the ability to find himself in such a world (I'm leaning toward 'no, if it's possible now, it should still be possible'). If hypercomputation was actually possible that would mean that strong variant of CT would be false, because there would be something effectively computable that wasn't computable by a Turing machine. OK. In a way, that strong form of CT might already be false with comp, only in the 1p sense as you get a free random oracle as well as always staying consistent(and 'alive'), but it's not false in the 3p
Re: Question about PA and 1p
Hi Bruno, On 1/12/2012 1:01 PM, Bruno Marchal wrote: On 11 Jan 2012, at 19:35, acw wrote: On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. You can ascribe a sort of local consciousness to the person living, relatively to you, that Sigma_1 truth, but the person itself is really related to all the proofs (in Platonia) of that sentences (roughly speaking). OK, but that requires that I have a justification for a belief in Platonia. The closest that I can get to Platonia is something like the class of all verified proofs (which supervenes on some form of physical process.) I simply cannot see how Sigma_1 sentences can interface with each other such that one can know anything about another absent some form of physicality. If I take away all forms of physical means of communicating ideas, no chalkboards, paper, computer screens, etc., how can ideas be possibly communicated? Mere existence does not specify properties. I go so far as considering that the wavefunction and its unitary evolution exists and it is a sufficiently universal physical process to implement the UD, but the UD as just the equivalent to Integers, nay, that I cannot believe in. “One cannot speak about whatever one cannot talk.” ~ Maturana (1978, p. 49) Onward! Stephen Bruno I think you might be confusing structures/relations which can be contained within PA with PA itself. On 1/11/2012 12:07 PM, acw wrote: On 1/10/2012 17:48, Bruno Marchal wrote: On 10 Jan 2012, at 12:58, acw wrote: On 1/10/2012 12:03, Bruno Marchal wrote: On 09 Jan 2012, at 19:36, acw wrote: To put it more simply: if Church Turing Thesis(CTT) is correct, mathematics is the same for any system or being you can imagine. I am not sure why. Sigma_1 arithmetic would be the same; but higher mathematics (set theory, analysis) might still be different. If it's wrong, maybe stuff like concrete infinities, hypercomputation and infinite minds could exist and that would falsify COMP, however there is zero evidence for any of that being possible. Not sure, if CT is wrong, there would be finite machines, working in finite time, with well defined instructions, which would be NOT Turing emulable. Hypercomputation and infinite (human) minds would contradict comp, not CT. On the contrary, they need CT to claim that they compute more than any programmable machines. CT is part of comp, but comp is not part of CT. Beyond this, I agree with your reply to Craig. In that response I was using CT in the more unrestricted form: all effectively computable functions are Turing-computable. I understand, but that is confusing. David Deutsch and many physicists are a bit responsible of that confusion, by attempting to have a notion of effectivity relying on physics. The original statement of Church, Turing, Markov, Post, ... concerns only the intuitively human computable functions, or the functions computable by finitary means. It asserts that the class of such intuitively computable functions is the same as the class of functions computable by some Turing machine (or by the unique universal Turing machine). Such a notion is a priori completely independent of the notion of computable by physical means. Yes, with the usual notion of Turing-computable, you don't really need more than arithmetic. It might be a bit stronger than the usual equivalency proofs between a very wide range of models of computation (Turing machines, Abacus/PA machines, (primitive) recursive functions (+minimization), all kinds of more current models of computation, languages and so on). Yes. I even suspect that CT makes the class of functions
Re: Question about PA and 1p
Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen On 1/11/2012 12:07 PM, acw wrote: On 1/10/2012 17:48, Bruno Marchal wrote: On 10 Jan 2012, at 12:58, acw wrote: On 1/10/2012 12:03, Bruno Marchal wrote: On 09 Jan 2012, at 19:36, acw wrote: To put it more simply: if Church Turing Thesis(CTT) is correct, mathematics is the same for any system or being you can imagine. I am not sure why. Sigma_1 arithmetic would be the same; but higher mathematics (set theory, analysis) might still be different. If it's wrong, maybe stuff like concrete infinities, hypercomputation and infinite minds could exist and that would falsify COMP, however there is zero evidence for any of that being possible. Not sure, if CT is wrong, there would be finite machines, working in finite time, with well defined instructions, which would be NOT Turing emulable. Hypercomputation and infinite (human) minds would contradict comp, not CT. On the contrary, they need CT to claim that they compute more than any programmable machines. CT is part of comp, but comp is not part of CT. Beyond this, I agree with your reply to Craig. In that response I was using CT in the more unrestricted form: all effectively computable functions are Turing-computable. I understand, but that is confusing. David Deutsch and many physicists are a bit responsible of that confusion, by attempting to have a notion of effectivity relying on physics. The original statement of Church, Turing, Markov, Post, ... concerns only the intuitively human computable functions, or the functions computable by finitary means. It asserts that the class of such intuitively computable functions is the same as the class of functions computable by some Turing machine (or by the unique universal Turing machine). Such a notion is a priori completely independent of the notion of computable by physical means. Yes, with the usual notion of Turing-computable, you don't really need more than arithmetic. It might be a bit stronger than the usual equivalency proofs between a very wide range of models of computation (Turing machines, Abacus/PA machines, (primitive) recursive functions (+minimization), all kinds of more current models of computation, languages and so on). Yes. I even suspect that CT makes the class of functions computable by physics greater than the class of Church. That could be possible, but more evidence is needed for this(beyond the random oracle). I also wonder 2 other things: 1) would we be able to really know if we find ourselves in such a world (I'm leaning toward unlikely, but I'm agnostic about this) 2) would someone performing my experiment(described in another message), lose the ability to find himself in such a world (I'm leaning toward 'no, if it's possible now, it should still be possible'). If hypercomputation was actually possible that would mean that strong variant of CT would be false, because there would be something effectively computable that wasn't computable by a Turing machine. OK. In a way, that strong form of CT might already be false with comp, only in the 1p sense as you get a free random oracle as well as always staying consistent(and 'alive'), but it's not false in the 3p view... Yes. Comp makes physics a first person plural reality, and a priori we might be able to exploit the first plural indeterminacy to compute more function, like we know already that we have more processes, like that free random oracle. The empirical fact that quantum computer does not violate CT can make us doubt about this. In the third person, there's no need to consider more than UD, which seems to place some limits on what is possible, but in the first person, the possibilities are more plentiful (if COMP). Also, I do wonder if the same universality that is present in the current CT would be present in hypercomputation (if one were to assume it would be possible) Yes, at least for many type of hypercomputation, notably of the form of computability with some oracle. - would it even retain CT's current immunity from diagonalization? Yes. Actually the immunity of the class of computable functions entails the immunity of the class of computable functions with oracle. So the consistency of CT entails the consistency of some super-CT for larger class. But I doubt that there is a super-CT for the class of functions computable by physical means. I am a bit agnostic on that. OK, although this doesn't seem trivial to me. As for the mathematical truth part, I mostly meant that from the perspective of
Re: Question about PA and 1p
On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. I think you might be confusing structures/relations which can be contained within PA with PA itself. On 1/11/2012 12:07 PM, acw wrote: On 1/10/2012 17:48, Bruno Marchal wrote: On 10 Jan 2012, at 12:58, acw wrote: On 1/10/2012 12:03, Bruno Marchal wrote: On 09 Jan 2012, at 19:36, acw wrote: To put it more simply: if Church Turing Thesis(CTT) is correct, mathematics is the same for any system or being you can imagine. I am not sure why. Sigma_1 arithmetic would be the same; but higher mathematics (set theory, analysis) might still be different. If it's wrong, maybe stuff like concrete infinities, hypercomputation and infinite minds could exist and that would falsify COMP, however there is zero evidence for any of that being possible. Not sure, if CT is wrong, there would be finite machines, working in finite time, with well defined instructions, which would be NOT Turing emulable. Hypercomputation and infinite (human) minds would contradict comp, not CT. On the contrary, they need CT to claim that they compute more than any programmable machines. CT is part of comp, but comp is not part of CT. Beyond this, I agree with your reply to Craig. In that response I was using CT in the more unrestricted form: all effectively computable functions are Turing-computable. I understand, but that is confusing. David Deutsch and many physicists are a bit responsible of that confusion, by attempting to have a notion of effectivity relying on physics. The original statement of Church, Turing, Markov, Post, ... concerns only the intuitively human computable functions, or the functions computable by finitary means. It asserts that the class of such intuitively computable functions is the same as the class of functions computable by some Turing machine (or by the unique universal Turing machine). Such a notion is a priori completely independent of the notion of computable by physical means. Yes, with the usual notion of Turing-computable, you don't really need more than arithmetic. It might be a bit stronger than the usual equivalency proofs between a very wide range of models of computation (Turing machines, Abacus/PA machines, (primitive) recursive functions (+minimization), all kinds of more current models of computation, languages and so on). Yes. I even suspect that CT makes the class of functions computable by physics greater than the class of Church. That could be possible, but more evidence is needed for this(beyond the random oracle). I also wonder 2 other things: 1) would we be able to really know if we find ourselves in such a world (I'm leaning toward unlikely, but I'm agnostic about this) 2) would someone performing my experiment(described in another message), lose the ability to find himself in such a world (I'm leaning toward 'no, if it's possible now, it should still be possible'). If hypercomputation was actually possible that would mean that strong variant of CT would be false, because there would be something effectively computable that wasn't computable by a Turing machine. OK. In a way, that strong form of CT might already be false with comp, only in the 1p sense as you get a free random oracle as well as always staying consistent(and 'alive'), but it's not false in the 3p view... Yes. Comp makes physics a first person plural reality, and a priori we might be able to exploit the first plural indeterminacy to compute more function, like we know already that we have more processes, like that free random oracle. The empirical fact that quantum computer does not violate CT can make us doubt about this. In the
Re: Question about PA and 1p
Hi Acw, On 1/11/2012 1:35 PM, acw wrote: On 1/11/2012 19:22, Stephen P. King wrote: Hi, I have a question. Does not the Tennenbaum Theorem prevent the concept of first person plural from having a coherent meaning, since it seems to makes PA unique and singular? In other words, how can multiple copies of PA generate a plurality of first person since they would be an equivalence class. It seems to me that the concept of plurality of 1p requires a 3p to be coherent, but how does a 3p exist unless it is a 1p in the PA sense? Onward! Stephen My understanding of 1p plural is merely many 1p's sharing an apparent 3p world. OK, we could see this as an equivalence class of sorts where the relation between the 1p is a 4-diffeomorphism. The correspondence between frames of reference/coordinate systems and 1p's makes sense, but what defines its closure and compactness? There has to be something that requires the set to be finite. The demand that any one of the 1p in the set be representable as a recursively countable string might to the trick, but each must be recursively countable in some way. I think that there is a to do this and not violate the Tennenbaum theorem http://web.mat.bham.ac.uk/R.W.Kaye/papers/tennenbaum/tennenbaum.xhtml. I have an idea but do not know how to representing formally yet. That 3p world may or may not be globally coherent (it is most certainly locally coherent), and may or may not be computable, typically I imagine it as being locally computed by an infinity of TMs, from the 1p. If we accept Bruno's result then a 3p world must supervene on an infinite number of computations. I strongly suspect that there must be an infinity of 3p's, each a globally maximally coherent set of 1p's... At least one coherent 3p foundation exists as the UD, but that's something very different from the universe a structural realist would believe in (for example, 'this universe', or the MWI multiverse). So a coherent 3p foundation always exists, possibly an infinity of them. The parts (or even the whole) of the 3p foundation should be found within the UD. It seems to me that there cannot be just one 3p as it could not be finite. Consider the number of Boolean Algebras that we can map (via endomorphism?) to a single orthocomplete lattice, and that would be just for one quantum mechanical system. Each of the BA would be the representation of a 1p, maybe... I am not sure... As for PA's consciousness, I don't know, maybe Bruno can say a lot more about this. My understanding of consciousness in Bruno's theory is that an OM(Observer Moment) corresponds to a Sigma-1 sentence. I think you might be confusing structures/relations which can be contained within PA with PA itself. Here I part with Bruno as I do not think that a Sigma_1 sentence alone has the necessary and sufficient structure for consciousness to obtain. We can certainly see that the OMs - Sigma_1 but there is more involved that the mere content of an experience. We have to reproduce the *appearance* of the Cartesian theater effect to have consciousness. Onward! Stephen -- 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.
