Another clueless wikipedia article
exama...@gmail.com
This article assumes that desktop computers cannot be turing machines.
Here we go again. Why is there no shortage of these fools who fail to
understand the simple explanations in the cinderella book?
exama...@gmail.com
am going to remove those uninformed assumptions that the tape of a
turing machine is an actually infinitely long resource. In fact, at any
moment, the turing machine has a finite ID, and thus the infinity
talked of is potential infinity (much like the infinity of Z)
Furthermore, the TM is a mathematical model. Every TM denotes a
particular computation. The author probably doesn't recognize that. The
potential infinity there is an idealization. It does not mean that
physical machines cannot be TMs!
If you followed that naive logic, just because you can define finite
state machines that would not fit into this universe, desktop pcs would
not even be finite state machines.
Think about that for a while. What do I mean by "Every TM denotes a
particular computation"?
I am hoping it is clear now, and now this bit about desktop pcs not
being TMs is goodbye kansas. Why on earth do you think we are studying
turing machines then?
The standard reason for the potentially infinite tape of the TM is to
model the fact that an ongoing computation may extend the tape
(allocate more memory). This is usually thought to be a property of
desktop PCs as well. With some effort, it is possible to extend the PC
with as much memory as you want. There are no hard limits (except
physical limits which apply to EVERYTHING including FSMs). Did you ever
connect two PCs with a network? Fine. Then you know what this means.
For the record, all of this can be shown rigorously.
If you followed that naive logic, just because you can define finite
state machines that would not fit into this universe, desktop pcs would
not even be finite state machines.
Think about that for a while. What do I mean by "Every TM denotes a
particular computation"?
I am hoping it is clear now, and now this crap about desktop pcs not
being TMs is goodbye kansas. Why on earth do you think we are studying
turing machines then!!???
Ben Rudiak-Gould
-- Ben
Simon G Best
Certainly it strangely refers to having "previously stated that a
desktop PC, along with all real computers, are finite state machines",
even though they don't seem to have made such a statement earlier on in
the article. However, the statement that "a desktop PC, along with all
real computers, are finite state machines" is correct, and does not mean
that they are assuming "that desktop computers cannot be turing
machines." But their statement does imply that there are some Turing
Machines that a PC isn't, and can never be programmed to be, which is
correct.
Simon G Best
exama...@gmail.com
reasons. Please follow my response on the wiki.
Regards,
--
Eray
exama...@gmail.com
> Certainly it strangely refers to having "previously stated that a
> desktop PC, along with all real computers, are finite state machines",
> even though they don't seem to have made such a statement earlier on in
> the article. However, the statement that "a desktop PC, along with all
> real computers, are finite state machines" is correct, and does not mean
> that they are assuming "that desktop computers cannot be turing
> machines." But their statement does imply that there are some Turing
> Machines that a PC isn't, and can never be programmed to be, which is
> correct.
Yes, that's what I would agree with in the discussion with Ben, but I
have a problem with this. What sense does it make to interject this
discussion in an article about computability theory? Isn't the author
supposed to talk about partially recursive functions and what it means
for a number to be computable, what it means to be a function to be
computable, etc.? I think this article isn't the right place for
analyzing the differences and similarities between desktop PCs and
FSMs. I would appreciate a separate article dealing with this problem,
but I am afraid it would be hard to do favor to this subject, which is
somewhat interesting philosophically, but has little use for a computer
scientist (I think).
And sorry for getting angry in the original post. An argument that
"personal computers are not Turing Machines, they are only finite state
machines" was often brought up in this newsgroup, and I thought it was
an extension of that discussion. At any rate, there are also Finite
State Machines that a PC isn't, but that is not a very interesting
fact. I stated how I think that should be put in my response: A
desktop PC that is not extensible can be seen as either a FSM or a TM.
It is interesting to note that such a machine cannot implement a
universal Turing machine which could simulate any possible Turing
Machine
The problem with the current state of the article is, it at least
implies to the reader that you can't study desktop PCs by way of Turing
Machines, which is just plain wrong. That's why I think it is
misleading. Such suggestions seem to detriment from the importance of
the definition of a Turing Machine. The definition lies at the heart of
time/space complexity analysis which lets us analyze real algorithms
that run on real desktop PCs. This is more than just being "useful", I
think it is necessary to treat the PCs as TMs to make this work!
Second, the assertion that "all real computers are FSMs" seems
philosophically unsatisfactory. Is a growable cellular automata a
finite state machine? I think it's much better to treat it as a
full-fledged computer. You cannot expect the reader to infer all the
discussion that we are making by himself, and that first sentence about
desktop PCs seems irrelevant and confusing. The author would IMO have
to stress that by desktop PC he means a device that cannot be extended
in any way, has a fixed RAM chip, and no connection to the outside
world, etc. Why does not the author talk about the soda vending
machine which would be much much more appropriate as a positive
example?
But I think, eventually, there is no reason wikipedia should not be a
great reference for all basic computer science :)
Perhaps I'm just nitpicking. It may be sufficient just to change that
first sentence so it gets the reference right, and perhaps remove the
bit about "all real computers", I really think that bit is problematic.
Because it does imply that all real computers are FSMs and not TMs. Ben
also seems to have understood it like that and not in the correct sense
that you understood, so I think it is clear that the explanation there
leads to ambiguity!
Best Regards,
--
Eray
Simon G Best
> Simon G Best wrote:
>
> [...] What sense does it make to interject this
> discussion in an article about computability theory? Isn't the author
> supposed to talk about partially recursive functions and what it means
> for a number to be computable, what it means to be a function to be
> computable, etc.? I think this article isn't the right place for
> analyzing the differences and similarities between desktop PCs and
I think you're reading much more into that brief mention of PCs than is
really there, or was ever intended. It looks to me much more like an
illustrative example of the limits of the computational power of FSMs -
entirely on-topic!
> The problem with the current state of the article is, it at least
> implies to the reader that you can't study desktop PCs by way of Turing
> Machines, which is just plain wrong. [...]
It does not imply any such thing.
> [...]
> Second, the assertion that "all real computers are FSMs" seems
> philosophically unsatisfactory. [...] The author would IMO have
> to stress that by desktop PC he means a device that cannot be extended
> in any way, has a fixed RAM chip, and no connection to the outside
> world, etc. [...]
A PC is not the same as a PC combined with an inexhaustible supply of
extra memory with the means to add that extra memory as and when needed.
A PC itself is a finite state machine. The fact that it could be
repeatedly extended into ever larger finite state machines, with more
and more states each time, is irrelevant to the fact that it is a finite
state machine. There was no need for the author to point out that they
don't mean
PCs-combined-with-infinitely-large-warehouses-of-extra-memory, because
no one would sensibly confuse the term "desktop PC" with such a concept.
> Perhaps I'm just nitpicking.
Nitpicking? You're criticising the article for saying things it isn't
even saying!
> It may be sufficient just to change that
> first sentence so it gets the reference right, and perhaps remove the
> bit about "all real computers", I really think that bit is problematic.
> Because it does imply that all real computers are FSMs and not TMs. [...]
No. That may be what you're (incorrectly) inferring from it, but it's
not what it's saying or implying.
:-(
Simon
exama...@gmail.com
> > It may be sufficient just to change that
> > first sentence so it gets the reference right, and perhaps remove the
> > bit about "all real computers", I really think that bit is problematic.
> > Because it does imply that all real computers are FSMs and not TMs. [...]
>
> No. That may be what you're (incorrectly) inferring from it, but it's
> not what it's saying or implying.
>
> :-(
Hmm, Ben seemed to draw the same thing as I did, and he even agreed to
that case (saying that physical computers cannot implement countable
infinity). That's why I now think it is misleading. It is an ambiguous
expression there.
Anyway, I agree with your view that a real computer can be
both an FSM and a TM, but I don't think that a reader will
easily notice that unless he already knows it. Most likely, he
will think that TMs are not about desktop computers!
Regards,
--
Eray
Simon G Best
>
> Hmm, Ben seemed to draw the same thing as I did, and he even agreed to
> that case (saying that physical computers cannot implement countable
> infinity). That's why I now think it is misleading. It is an ambiguous
> expression there.
I'm not convinced. He was responding to what you said the article said,
but what you said the article said was not actually what the article
said. So, if he's got the impression that the article said that PCs
aren't Turing Machines, is it because he got that impression from the
article itself, or because you've given him that impression? (He might
have just accepted what you incorrectly stated about what the article
said, with the result that he's got the wrong impression of it as a
result. I've seen that sort of thing happen before.)
It's also possible that there's also confusion here about what can be
meant by such expressions as 'PCs aren't Turing Machines'. Is it
'Turing Machines' as in the machines themselves? Or is it 'Turing
Machines' in the sense of the model of computing represented by Turing
Machines collectively and generally? Sometimes we mean the latter, in
which case we're not saying that there is no Turing Machine that a PC is
(or is equivalent to), but that no PC is a Universal Turing Machine (or
equivalent). I think that's the sense in which Ben means it.
But there's also a red-herring case, as well, which may be relevant.
Sometimes, people point out that Turing Machines have inexhaustible
storage space on their tapes, which PCs do not have. Therefore, they
claim, PCs cannot be Turing Machines (in the sense of the machines
themselves), no matter what Turing Machine we might be considering. But
this is confusing what we could call 'implementation details' with the
computations represented by those machines. For example, consider a
Turing Machine which immediately halts, no matter what its input is.
The output will be identical to the input. Such a Turing Machine can be
implemented on a PC by just immediately echoing everything entered
through the keyboard (the input) straight to the screen (the output),
key-press by key-press, in a never-ending loop - the output will be
identical to the input, no matter how much input there is!
Simon
exama...@gmail.com
You have my deepest respect. An excellent explanation.
The fact that a turing machine is just a model is often overlooked. In
the article, I tend to think that a clear example of a device that can
be only considered to be a FSM and not a more powerful computer would
be much better. Why, because it automatically avoids such long-winded
thoughts as these. Thus, while the FSM-TM-Desktop PC discussion may be
useful, it would better be moved as additional material. Why skip the
soda vending machine, which is a much more natural example?
By the way, what is such a natural example of PDA?
Regards,
--
Eray
Simon G Best
Thank you :-)
> The fact that a turing machine is just a model is often overlooked. In
> the article, I tend to think that a clear example of a device that can
> be only considered to be a FSM and not a more powerful computer would
> be much better.
But a PC is not "a more powerful computer" than an FSM. And all FSMs
are Turing Machines (or equivalents) anyway, so there are no examples of
FSMs "that can be only considered to be a FSM" and not also a Turing
Machine.
> Why, because it automatically avoids such long-winded
> thoughts as these.
Actually, I think it's a rather good thing if it intrigues readers and
inspires them to actually think about such things, discuss and discover,
etc :-)
> Thus, while the FSM-TM-Desktop PC discussion may be
> useful, it would better be moved as additional material. Why skip the
> soda vending machine, which is a much more natural example?
All soda vending machines are Turing Machines (or equivalents) (but not
all Turing Machines are soda vending machines, of course).
> By the way, what is such a natural example of PDA?
A bureaucrat?
Simon
Marissa Kay
http://en.wikipedia.org/wiki/Computability_theory_%28computer_science%29
examach...@gmail.com wrote:
> This article assumes that desktop computers cannot be turing machines.
[...]
> It does not mean that physical machines cannot be TMs!
Neither of these, however, have anything to do with what the article
said.
All finite state machines, and therefore all physical systems, are
representable as Turing Machines. This is also what the article implies
and states -- exactly the opposite of what you're describing above.
All physical systems are finite state machines (more precisely, finite
state Markov processes), the number of states being given by the
exponential of the entropy (in units of Boltzmann's constant) of the
system (which is, in fact, how entropy is defined); and the state
transition function being that generated by the system's Hamiltonian.
The Bekenstein Bound, in turn, places a hard limit on the size of the
state space and on the entropy of anything occupying a given region of
space (and even the space, itself), no matter what its constituency.
More on that, below.
At the microscopic level the state transition is deterministic --
regardless of whether one's talking about classical or quantum physics
or any of the classico-quantum hybrids comprising quantum theories with
superselection. Non-determinism (be it in classical or quantum physics,
or classico-quantum hybridge) is only arrived at by coarse-grained
representations of systems.
The finitude of the state space is, in fact, a centrally important
feature of statistical and thermal physics, associated with the notions
of recurrence and ergodicity. Since the microscopic system is
deterministic, then it cycles through its state space in a finite
amount of time. Since the Hamiltionian generates a non-dispersive state
transition, this actually divides the state space into separate
sectors, each sector comprising the closure of all the states that will
be cycled through from a given state.
This relates directly to the concept of ergodicity, and it is
ergodicity, itself, that gives the primary justification for the whole
process of providing coarse-grained non-deterministic representations
at the macroscopic level for underlying deterministic systems.
For quantum systems, both concepts of recurrence/recurrence sectors and
ergodicity are closely related to the concept of superselection, which
was alluded to above. A quantum system is ergodic if and only if no
observable (other than the Hamiltonian) commutes with the Hamiltonian.
A system with superselection has observables that commute with
everything.
There is, in fact, a standard notion of "state", couched within the
language of the "algebraic approach" to classical, statistical and
quantum physics initiated by no less than same von Neumann, himself,
who brought about the von Neumann machine concept that is central to
modern computer engineering. The close tie-in between computer science
and physics in all the foregoing is an imprint marking the presence of
von Neumann.
No machine, at the present time, uses more than a tiny fraction of the
(finite) state space available to it, nor will any in the foreseeable
future possibly until the full development of nanotechnology enables
the employment of the atomic and subatomic structure of the physical
system in question.
All systems occupying a finite volume -- including space, itself, and
the fields pervading it -- no matter what their composition, will be
subject to the Bekenstein Bound.
The Bekenstein Bound, more precisely, provides a hard-limit on the size
of the state space associated with anything in any region of space
bounded by a surface of area A. It is the exponential of A, itself,
with A given in units of 1/4 of the Planck area. The hard-limit on the
system's entropy is kA, where k is Boltzmann's constant and A, again,
in units of 1/4 the Planck area.
In fact, Jacobson in 1995 showed that the law of gravity (as given by
Einstein's equation in General Relativity) can be DERIVED from the
existence of a hard limit of this nature. The Raychaudhuri equation
figures crucially, in the derivation, in generating a kind of
"gravitational lensing" effect that's required to get Einstein's
equations.
He showed not only that the existence of a hard limit to entropy
proportional to area, when taken in conjunction with the 1st and 2nd
laws of thermodynamics, yields Einstein's equations in General
Relativity, but that the unit of area involved in the proportion has to
be 1/4 the Planck area -- thus deriving the Bekenstein Bound, itself.
It turns out that for a spherical region, the limit is exactly that
which is reached the moment enough information and matter is crammed
into the region to produce a black hole of the same size as the region,
itself. The theoretical maximum entropy of the region is then realized
as none other than what has been asserted since the 1970's to be the
entropy of the black hole, itself.
So, to make a long story short, the foregoing stands as a decisive
counter-point to what you were (apparently) trying to say. If you cram
too much storage, fields, light, electricity or matter (or even sound,
pressure, tension, stress or energy) into a given machine, you'll
eventually reach the point when the whole thing will collapse under its
own gravity and become a black hole. At that point, the theoretical
maximum will be reached for the region enclosed by the black hole.
Anything more crammed in will just make the black hole larger.
(But you'll never see it get larger, unless you fall in, or see
anything that's too close to the event horizon at all, since time's
nearly frozen to a standstill on the event horizon and everything close
to it is red shifted into invisibility).
Eventually, as per Hawking, the black hole will eventually radiate all
of its energy away (and all of its information contents) as thermal
radiation and evaporate away.
Interestingly, there actually was a report from Brookhaven last year,
that made the news, that a microscopic black hole might have come
momentarily into existence before evaporating, though I don't know if
it was ever confirmed. There was something peculiar about the pattern
in the decay products, decay times or something, that seemed to mark
the signature of a momentary black hole.
You mentioned, by the way, something about the universe, itself, then
being nothing more than a FSM if all of its parts were to be. That does
not follow, not even with the fact that all the finite-volumes within
the universe have finite state spaces.
That is, it does not follow, unless the Universe, itself, is finite.
But in either case, it still doesn't even follow that the Universe ANY
well-defined state space at all, in the first place, finite or
infinite. As Smolin has pointed out, since the underlying state space
(as per Quantum Gravity) is a certain space of 3-manifolds, and since
the classification problem for 3-manifolds may very well be
unsolveable, it's quite possible that there may not be any effectively
defineable state space for the Universe at all.
But the one thing that CAN be said is that the state space associated
with the SKY (i.e. the past light cone of any spacetime point) is most
likely finite. That's because the sky -- or the past light cone -- has
the structure of a hypersphere. That's true, regardless of whether the
Universe is finite with positive curvature, infinite and spatially flat
or infinite with negative curvature. In all cases, the sky is a
hypersphere that has the Big Bang at its antipode. The Big Bang is the
one and only event that is directly in the line of sight in every
direction from every place in the Unverse at every time (though it is
somewhat obscured by the cosmic microwave background or CMB). The
antipode and the CMB surrounding it appear as a giant spherical shell
some 12-14 billion light years in radius, but is actually a small shell
that is concave the other way, with us on the outside, and with the
glow of the Big Bang inside it at its center.
I'm not entirely sure how the Bekenstein Bound works with this, since a
hypersphere is a closed surface and doesn't have a boundary. Maybe the
limit is that associated with the area of the largest spherical
cross-section of the past light cone.
To date, as far as I know, nobody's ever tried to formulate a synthesis
of quantum theory and general relativity making use of the state space
of the past light cone. It may overcome Smolin's objection.
Now, the classical non-trivial example of an infinite state machine
(ISM) is that recognizing the Dyck language, given by the grammar
S -> 1; S -> u S v S
(1 = empty word); which has as its minimal state diagram
[[0]] -- u --> [1] -- u --> [2] -- u --> [3] ...
<-- v -- <-- v -- <-- v --
with an infinite number of states -- easily recognized as the state
diagram of a particular 1-counter machine.
In theory, a quantum system like the electromagnetic field is supposed
to provide a similar infinity of states, each frequency mode admitting
a set of states containing 0, 1, 2, 3 or more quanta of that frequency.
However, this is known to have a breakdown at a high enough energy (as
does the rest of quantum fieid theory) -- certainly by the time you
bump into the Bekenstein Bound limit. The manner of breakdown, though,
is still unknown.
Jacobson alluded indirectly to the breakdown phenomenon, pointing out
that the quanta of gravitational interaction has a breakdown mode
analogous to that which occurs for phonons in solids -- which leads to
a notion of gravitons as "vacuum phonons".
Other than theoretical construct of the quantum field (which, as per
the above, breaks down), there is nothing that can embody even the
deceptively simple automaton depicted by the state diagram above, since
it has an infinite number of states. All the more so for any of the
varieties of other infinite state automata, including the Push Down
automata or Turing Machines.
There are, of course, instances of each of these classes where the
number of states that can be reached from the start state is finite.
Then, since the minimal state diagram is finite, they are equally
representable as finite state machines. But in the general case -- like
that for the Dyck language above -- the number of states accessible
from the start state is infinite, the minimal state diagram is infinite
and it is simply not representable as a finite state machine.
The utility of Turing Machines or any of the other varieties of
infinite state machine formalisms (push down automata, counter
automata, etc.) is that they provide a more efficient means of
representing finite state machines and the physical systems they may
embody. Obviously, the more powerful a formalism, the greater utility
it will have -- even for those things that could theoretically be
represented by lesser means.
Greater utility also means greater relevance -- which is exactly the
opposite of what you're arguing!
Oracles provide a more powerful mechanism, still, and despite the fact
that they are strictly more powerful than TM's, they are neither no
less of an idealization than any other infinite state automaton
formalism, nor any less physically relevant. But theorists haven't yet
made a great deal of use of them. There is, just as well, a notion of
resource bounded oracles, for each variety of oracle, and this is bound
to be just as relevant for finite state machines and physical systems
as is that of the resource bounded TM.
One example of an application of the oracle concept was in the
high-level description of the "Mathematician's Algorithm" (provided
several years ago in sci.math.research, as can be found by doing a
Google Groups search), which is an enveloping framework for processes
that embody the activity that appears to occur during the course of
developing mathematical formalisms, finding and proving conjectures,
etc. Though it may not be precisely emulatable in any physical
mechanism (but then, so what? Neither is the infinite state automaton
depicted above or a universal TM), it is highly relevant, providing the
basis for a nested series of approximate relaizations that may seek to
partially automate more and more of the process described.
The primary mode of application of any of the varieties of ISM is
through the notion of "resource-boundedness". A resource bounded ISM is
just one with its state space truncated in some way. All
resource-bounded TM's are finite state machines; and it's primarily
through this mode of application that TM's have their greatest
relevance.
The whole foundation of complexity theory, itself, is based on this
notion of resource boundedness.
exama...@gmail.com
tidbit here was that an article examining the physical realizability of
certain classes of machines would be a nice addition to wikipedia, but
I don't like the examples in the computability article.
Best,
mark...@yahoo.com
Take a look at Baez's article and the adjoining thread in
sci.physics.research on the physical realizability of P = NP.
The issue of the interrelation between Physics and Computer Science is,
in fact, quite active these days, and too much time has gone by with it
being rebuffed by both fields. That's pretty much the point the papers
listed in the references cited in the articles on that thread make, as
well.
It does put the ball in your court. For, now, if you want the machine
to be omni-expandable, it's going to have to be endowed with an
indefinite capacity to physically grow larger and larger.
But then, with all these billions of years and all the galaxies and
star systems out there, you'd expect to have seen it already done
somewhere else at least once already. That, of course, is the
punchline. If it gets big enough (like, as big as a galaxy), it'll be
very easy to see it from Earth. But nothing of the sort has ever been
seen anywhere out there. So, it's probably never been done ... and
probably never will be.
Do a Web search on "Omegon".
exama...@gmail.com
Thanks for your comments.
mark...@yahoo.com wrote:
> exama...@gmail.com wrote:
> > Thank you Marissa for this informative message. Much appreciated. My
> > tidbit here was that an article examining the physical realizability of
> > certain classes of machines would be a nice addition to wikipedia, but
> > I don't like the examples in the computability article.
>
> Take a look at Baez's article and the adjoining thread in
> sci.physics.research on the physical realizability of P = NP.
Interesting! Surely that is a physical question, no?
> The issue of the interrelation between Physics and Computer Science is,
> in fact, quite active these days, and too much time has gone by with it
> being rebuffed by both fields. That's pretty much the point the papers
> listed in the references cited in the articles on that thread make, as
> well.
That is interesting, thank you.
> It does put the ball in your court. For, now, if you want the machine
> to be omni-expandable, it's going to have to be endowed with an
> indefinite capacity to physically grow larger and larger.
I see :) That is kind of hard in a finite world.
For the record, the black-hole guys must consider that there is always
a slow, but large machine that doesn't collapse (right?) Even I could
prove that, no? :)
> But then, with all these billions of years and all the galaxies and
> star systems out there, you'd expect to have seen it already done
> somewhere else at least once already. That, of course, is the
> punchline. If it gets big enough (like, as big as a galaxy), it'll be
> very easy to see it from Earth. But nothing of the sort has ever been
> seen anywhere out there. So, it's probably never been done ... and
> probably never will be.
Well, since quantum events inherently support quantum computers, the
universe is a computer that is as large as it gets :) And since nothing
could be bigger, maybe it doesn't make sense to consider bigger
machines?
> Do a Web search on "Omegon".
I will, thank you.
Best Regards,
--
Eray
mark...@yahoo.com
> Well, since quantum events inherently support quantum computers, the
> universe is a computer that is as large as it gets :) And since nothing
> could be bigger, maybe it doesn't make sense to consider bigger
> machines?
The issue of simulating quantum systems by other quantum systems is
also pretty active, these days. I can't give you definite references,
but you might find something under "quantum simulation".
> > Do a Web search on "Omegon".
> I will, thank you.
Better is "Omega Point". It seems Omegon's been crowded in Web space by
some company that's decided to take up the name. You other speculations
were getting borderline Transhumanism -- they particularly like the
notion of the Singularity.
Take a look at John Baez's blog page. I think he still has something up
there about the Singularity idea, which has been bandied about from
time to time by logicians and physicists.
exama...@gmail.com
> exama...@gmail.com wrote:
> > Well, since quantum events inherently support quantum computers, the
> > universe is a computer that is as large as it gets :) And since nothing
> > could be bigger, maybe it doesn't make sense to consider bigger
> > machines?
>
> The issue of simulating quantum systems by other quantum systems is
> also pretty active, these days. I can't give you definite references,
> but you might find something under "quantum simulation".
I'd seen a paper on arxiv, yes. I think it is acceptable (intuitively)
that a quantum computer can simulate any physical system.
> > > Do a Web search on "Omegon".
> > I will, thank you.
>
> Better is "Omega Point". It seems Omegon's been crowded in Web space by
> some company that's decided to take up the name. You other speculations
> were getting borderline Transhumanism -- they particularly like the
> notion of the Singularity.
>
> Take a look at John Baez's blog page. I think he still has something up
> there about the Singularity idea, which has been bandied about from
> time to time by logicians and physicists.
Borderline transhumanism, yes. :) Chaitin has a very similar idea, in
an old
paper that I read he says that it may be possible to regard the
approximation of the halting probability (Omega) as an abstract model
of
all evolution on earth.
Interesting parallels with Chardin's much earlier idea. Thanks for the
comments again.
Best Regards,
---
Eray