### Re: Questions about the Equivalence Principle (EP) and GR

```

On Saturday, March 2, 2019 at 2:29:50 AM UTC-7, agrays...@gmail.com wrote:
>
>
>
> On Friday, March 1, 2019 at 10:14:02 PM UTC-7, agrays...@gmail.com wrote:
>>
>>
>>
>> On Thursday, February 28, 2019 at 12:09:27 PM UTC-7, Brent wrote:
>>>
>>>
>>>
>>> On 2/28/2019 4:07 AM, agrays...@gmail.com wrote:
>>>
>>>
>>>
>>> On Wednesday, February 27, 2019 at 8:10:16 PM UTC-7, Brent wrote:

On 2/27/2019 4:58 PM, agrays...@gmail.com wrote:

*Are you assuming uniqueness to tensors; that only tensors can produce
covariance in 4-space? Is that established or a mathematical speculation?
TIA, AG *

That's looking at it the wrong way around.  Anything that transforms as
an object in space, must be representable by tensors. The informal
definition of a tensor is something that transforms like an object, i.e.
in
three space it's something that has a location and an orientation and
three
extensions.  Something that doesn't transform as a tensor under coordinate
system changes is something that depends on the arbitrary choice of
coordinate system and so cannot be a fundamental physical object.

Brent

>>>
>>> 1) Is it correct to say that tensors in E's field equations can be
>>> represented as 4x4 matrices which have different representations depending
>>> on the coordinate system being used, but represent the same object?
>>>
>>>
>>> That's right as far as it goes.   Tensors can be of any order.  The
>>> curvature tensor is 4x4x4x4.
>>>
>>> 2) In SR we use the LT to transform from one* non-accelerating* frame
>>> to another. In GR, what is the transformation for going from one
>>> *accelerating* frame to another?
>>>
>>>
>>> The Lorentz transform, but only in a local patch.
>>>
>>
>> *That's what I thought you would say. But how does this advance
>> Einstein's presumed project of finding how the laws of physics are
>> invariant for accelerating frames? How did it morph into a theory of
>> gravity? TIA, AG *
>>
>
> *Or suppose, using GR, that two frames are NOT within the same local
> patch.  If we can't use the LT, how can we transform from one frame to the
> other? TIA, AG *
>
> *Or suppose we have two arbitrary accelerating frames, again NOT within
> the same local patch, is it true that Maxwell's Equations are covariant
> under some transformation, and what is that transformation? TIA, AG*
>

*I think I can simplify my issue here, if indeed there is an issue: did
Einstein, or anyone, ever prove what I will call the General Principle of
Relativity, namely that the laws of physics are invariant for accelerating
frames? If the answer is affirmative, is there a transformation equation
for Maxwell's Equations which leaves them unchanged for arbitrary
accelerating frames? TIA, AG *

>
>>> Brent
>>>
>>

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```

### Re: When Did Consciousness Begin?

```

On 3/5/2019 4:06 AM, Bruno Marchal wrote:

RNA, proteins, krebs cycle, and proton pumps are all necessary for that.

That is carbon chauvinism, with all my respect. I am a lover of Krebs
cycle, and even more Calvin cycle (in photosynthesis). My initial
inspiration of Mechanism came from Molecular biology. But nothing
there has been shown to be non-Turing emulable. Your artificial brain,
when you say “yes” to the doctor, might not involve any of these
cycles, but use a simple battery instead (or you are just telling me
that you doubt Digital Mechanism, which is my basic working hypothesis
to solve the Mind-Body problem.

That you can emulate those processes is beside the point.  The point is
that you would have to emulate them in order to support your contention
that bacteria are Turing complete.  That's has been my "doubt" of your
theory all along.  It is not a TOE in which consciousness appears
without matter.  It is a theory in which consciousness and matter must
appear together.  Every time I mention this you strike back at the straw
man of primitive matter...which I never refer to.

Brent

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### Re: Recommend this article, Even just for the Wheeler quote near the end

```

On Tuesday, March 5, 2019 at 6:23:42 AM UTC-6, Bruno Marchal wrote:
>
>
> On 5 Mar 2019, at 00:43, Brent Meeker >
> wrote:
>
>
>
> On 3/4/2019 3:54 AM, Bruno Marchal wrote:
>
>
> On 3 Mar 2019, at 20:43, Brent Meeker >
> wrote:
>
>
>
> On 3/3/2019 4:52 AM, Philip Thrift wrote:
>
>
>>
> Here's an example David Wallace presents (as an "outlandish" possibility):
> Suppose in *pi *(which is computable, so has a *program* (a spigot one,
> in fact) that produces its digits. Suppose somewhere in that stream of
> digits is the Standard Model Equation
>
> (say written in LaTeX/Math but rendered here)
>
>
> So what could this mean? (He sort of leaves it hanging.)
>
>
> Nothing.  Given a suitable mapping the SM Lagrangian can be found in any
> sequence of symbols.  It's just a special case of the rock that computes
> everything.
>
>
> Even if rock would exist in some primitive sense, which I doubt, they do
> not compute anything, except in a trivial sense the quantum state of the
> rock. A rock is not even a definable digital object.
>
>
> It's an ostensively definable object...which is much better.
>
>
> Ostension is dream-able.
>
>
>
>
>
> If someone want to convince me that a rock can compute everything, I will
> ask them to write a complier of the combinators, say, in the rock. I will
> ask an algorithm generating the phi_i associated to the rock.
>
>
> There is no particular phi_i associated to the rock.  That's the point.
> The rock goes thru various states so there exists a mapping from that
> sequence of states to any computation with a similar number of states.
>
>
> It is a mapping of states. It is like a bijection. You need something like
> a morphism preserving the computability structure, which do not exist in
> the rock. A computation is not just a sequence of states, it is a sequence
> of states defined by the universal machine which brought those states.
>
> There are bijections between N and Z, but only Z is a group, because those
> bijections does not preserve the algebraic structure. Similarly, there is a
> bijection between a computation and a movie of that computation, but it
> does not preserve the causal/logical relation between the states, which is
> a universal machine for the computation, and just a linear order for the
> sequence, without structure, of the states.
>
>
>
>   Of course one may object that the actual computation is in the
> mapping...but that's because of our prejudice for increasing entropy.
>
>
> OK.Now, a bijection between a physical computation and an arithmetical
> computation do preserve the computability structure, that is why we can say
> that the arithmetical reality/model implements genuinely the computations.
>
> Bruno
>
>
>

The bijection

material [physical] computation ↔ arithmetical computation

is like (New Testament) Paul's thesis: There's earthly bodies and spiritual
bodies.

"Not all flesh is the same: People have one kind of flesh, animals have
another, birds another and fish another. There are also heavenly bodies and
there are earthly bodies; but the splendor of the heavenly bodies is one
kind, and the splendor of the earthly bodies is another. ... If there is a
natural body, there is also a spiritual body."

Spiritual or heavenly fictionalism is like arithmetical fictionalism:
spirits (like numbers) do not exist.

- pt

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### Re: A Program to Compute Gödel-Löb Fixpoints

```
> On 2 Mar 2019, at 14:19, Lawrence Crowell
> wrote:
>
> I guess I am not sure what a Gödel-Löb fixed point is. Is this somehow
> analogous to a Brouwer fixed points in maps or diffeomorphisms of spaces?

There are some analogies, some of which can be made technically precise, but
that requires sophisticated semantics for the variant of probability.

The fixed point theorem in provability logic is that self-reference have fixed
point, like with the diagonalisation lemma.

By the diagonalisation lemma, you can find sentence asserting anything
computable about them, and you can eliminate the self-reference. For example,
you can find p such that PA proves p <-> []p, that gives the Löbian fixed point
t, and PA will prove it, PA will prove p <-> t.
Similarly PA can prove, for some p, that p <-> ~[]p, that gives the famous
Gödel sentences, and PA will prove that p is equivalent, in this case, to <>t
(= ~[]f = consistency).
There are many others, and even if they involved other sentences, it is always
possible to eliminate the self-reference.
Three different proofs are given in Boolos 1993. It is a very important result
in the provability logic, aka machine theology (to be sure the theology is G*,
the true extension of the provability logic G, coming from Solovay second
completeness theorem, which generalises a lot the incompleteness theorem.
Indeed all the true propositional modal truth are axiomatised by G*, also known
as GLS (Gödel-Löb-Solovay) in the literature.

>
> I read Rucker's Infinity and the Mind last spring, after having read it many
> years ago. I could tell he had a penchant for various mystical ideas. This
> tends blog entry of his suggests he has ideas similar to what Gödel thought,
> and which I think were a part of leading  him into paranoid delusions. When I
> clicked on this for some reason I thought this was about monoids, and was a
> bit disappointed to see it is more philosophical. However, I think the Kant
> noumena is not really directly knowable, and I think from quantum mechanics
> we can't know this as either purely epistemic or ontic. I am not sure how
> ideas of mind fit into this.

With mechanism, it fits remarkably. Indeed it makes physics a branch of
machine’s theology. See my papers for all details, or my posts here. Physics
can be sued to refute that theology, but we get only confirmation up to now.
The physicalist theory of mind requires endowing matter with some magics, for
which no evidence exist.

Bruno

>
> LC
>
> On Saturday, March 2, 2019 at 3:26:18 AM UTC-6, Philip Thrift wrote:
>
>
> A Program to Compute Gödel-Löb Fixpoints
> Melvin Fitting [ http://melvinfitting.org/  ]
>
> https://www.researchgate.net/publication/285841645_A_program_to_compute_Godel-Lob_fixpoints
>
>
>
>
> A loose motivation for much of Melvin Fitting's work can be formulated
> succinctly as follows. There are many logics. Our principles of reasoning
> vary with context and subject matter. Multiplicity is one of the glories of
> modern formal logic. The common thread tying logics together is a concern for
> what can be said (syntax), what that means (semantics), and relationships
> between the two. A philosophical position that can be embodied in a formal
> logic has been shown to be coherent, not correct. Logic is a tool, not a
> master, but it is an enjoyable tool to use.
> [ https://en.wikipedia.org/wiki/Melvin_Fitting
>  ]
>
>
> also (a bit offbeat):
>
> “Simply Gödel,” Phenomenology, and Monads
> Rudy Rucker
>
>
> - pt
>
> --
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### Re: When Did Consciousness Begin?

```
> On 2 Mar 2019, at 08:22, Philip Thrift  wrote:
>
>
>
> On Friday, March 1, 2019 at 6:25:05 PM UTC-6, Bruno Marchal wrote:
>
>> On 1 Mar 2019, at 09:28, Philip Thrift >
>> wrote:
>>
>>
>>
>> On Friday, March 1, 2019 at 2:05:03 AM UTC-6, Philip Thrift wrote:
>>
>>
>> On Thursday, February 28, 2019 at 5:15:17 PM UTC-6, Brent wrote:
>>
>>
>> On 2/28/2019 3:00 PM, Philip Thrift wrote:
>>>
>>>
>>> On Thursday, February 28, 2019 at 4:34:54 PM UTC-6, Brent wrote:
>>>
>>>
>>> On 2/28/2019 2:14 PM, Philip Thrift wrote:

On Thursday, February 28, 2019 at 3:48:04 PM UTC-6, Brent wrote:

On 2/28/2019 1:17 PM, Philip Thrift wrote:
>
>
> The best current philosopher of (and writer about) consciousness is Galen
> Strawson.
>
> https://en.wikipedia.org/wiki/Galen_Strawson
>
>
> https://liberalarts.utexas.edu/philosophy/faculty/profile.php?id=gs24429
>
>
> There is a lot of his material (PDFs, articles, videos, etc.) freely
> available online.
>
> The main word that is synonymous with consciousness is experience.

Which is something bacteria and plants and my thermostat have...and
ability to detect and react to the environment based on internal states.

Brent

Galen is a (type of) micropsychist.
>>>
>>> But the point is we don't need a philosopher to explain that level of
>>> consciousness to us.  It's already at the level of engineering.  If
>>> Strawson is going to provide any useful explanations of consciousness he
>>> should study machine learning...it's getting close to engineering
>>> consciousness at the next higher level.
>>>
>>> Brent
>>>
>>> It won't be accomplished via certain types of engineering, like
>>> "information network" approaches (IIT [
>>> https://en.wikipedia.org/wiki/Integrated_information_theory
>>>  ]) but
>>> potentially could with a "synthetic" approach that combines networks with
>>> synthetic biology. Something along these lines is the "fusion" idea
>>> proposed by
>>
>> I don't know why IIT is even discussed.  Aaronson pretty well shot it down.
>>
>> My son may get a chance to work on the Deepmind team.  What kind of brain
>> cells would you suggest he sprinkle on the CPUs?
>>
>> Brent
>>
>>
>> Like The Graduate's "plastics", today, "polymers".
>>
>>
>> Biomaterials for the central nervous system
>> https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2475552/
>>
>>
>> Scientists Have Built Artificial Neurons That Fully Mimic Human Brain Cells
>>
>>
>>
>> Scientists develop promising new type of polymer
>> https://phys.org/news/2019-01-scientists-polymers.html
>>
>>
>> Synthetic Glycopolymers for Highly Efficient Differentiation of Embryonic
>> Stem Cells into Neurons: Lipo- or Not?
>> https://www.ncbi.nlm.nih.gov/pubmed/28287262
>>
>>
>> Elastic materials for tissue engineering applications: Natural, synthetic,
>> and hybrid polymers
>> https://www.sciencedirect.com/science/article/pii/S174270611830494X
>>
>>
>> Biomaterials for Scaffolds: Synthetic Polymers
>> https://www.researchgate.net/publication/286340849_Biomaterials_for_Scaffolds_Synthetic_Polymers
>>
>>
>>
>>
>>
>> Biosynthetic Polymers as Functional Materials
>> https://pubs.acs.org/doi/full/10.1021/acs.macromol.6b00439
>>
>
>
> You might be interested by this quite remarkable news: a 8 letters synthetic
> DNA, which seems to work well.
> If that is true, it really suggests that we all come from one bacteria, I
> think. It is amazing that all life use only the same 4 letters coding (A, T,
> G, C).
>
> https://www.nature.com/articles/d41586-019-00650-8
>
>
> Bruno
>
>
> As this sort of stuff progresses, and bioengineers make conscious 'robots'
> out of alternative materials, phenomenologists will wonder how their
> experiences differ from ours.
>
> (But now Dan Dennett says Don't make conscious robots in the first place.)

The universal “virgin” machine is maximally conscious. Any non universal
```

### Re: Recommend this article, Even just for the Wheeler quote near the end

```
> On 1 Mar 2019, at 20:42, Philip Thrift  wrote:
>
>
>
> On Friday, March 1, 2019 at 8:49:54 AM UTC-6, Bruno Marchal wrote:
>
>> On 1 Mar 2019, at 01:42, Lawrence Crowell > > wrote:
>>
>>
>>
>> On Monday, February 25, 2019 at 9:42:01 AM UTC-6, Bruno Marchal wrote:
>>
>>> On 25 Feb 2019, at 12:39, Lawrence Crowell >
>>> wrote:
>>>
>>> On Monday, February 25, 2019 at 2:44:14 AM UTC-6, Bruno Marchal wrote:
>>>
On 24 Feb 2019, at 15:24, Lawrence Crowell >
wrote:

On Friday, February 22, 2019 at 3:18:01 PM UTC-6, Brent wrote:

On 2/22/2019 11:39 AM, Lawrence Crowell wrote:
> This sounds almost tautological. I have not read Masanes' paper, but he
> seems to be saying the Born rule is a matter of pure logic. In some ways
> that is what Born said.
>
> The Born rule is not hard to understand. If you have a state space with
> vectors |u_i> then a quantum state can be written as sum_ic_i|u_i>. For
> an observable O with eigenvectors o_i the expectation values for that
> observable is
>
>  sum_{ij} = sum_{ij} = sum_ip_io_i.
>
> So the expectations of each eigenvalue is multiple of the probability for
> the system to be found in that state. It is not hard to understand, but
> the problem is there is no general theorem and proof that the eigenvalues
> of an operator or observable are diagonal in the probabilities.
>>>
>>> I am not sure I understand this.
>>>
>>>
>>>
>>>
> In fact this has some subtle issues with degeneracies.

Doesn't Gleason's theorem show that there is no other consistent way to
assign probabilities to subspaces of a Hilbert space?

Brent

It is close. Gleason's theorem tells us that probabilities are a
consequence of certain measurements. So for a basis Q = {q_n} then in a
span in Q = P{q_n}, for P a projection operator that a measure μ(Q} is
given by a trace over projection operators. This is close, but it does not
address the issue of eigenvalues of an operator or observable. Gleason
tried to make this work for operators, but was ultimately not able to.
>>>
>>> It should work for the projection operator, that this is the
>>> yes-no-experiment, but that extends to the other measurement, by reducing
>>> (as usual) the question “what is the value of A” into the (many) question
>>> “does A measurement belong to this interval” … Gleason’s theorem assures
>>> that the measure is unique (on the subspaces of H with dim bigger or equal
>>> to 3), so the Born rule should be determined, at least in non degenerate
>>> case (but also in the degenerate case when the degeneracy is due to tracing
>>> out a subsystem from a bigger system. I will verify later as my mind
>>> belongs more to the combinator and applicative algebra that QM for now.
>>>
>>>
>>>

Many years ago I had an idea that since the trace of a density matrix may
be thought of as constructed from projection operators with tr(ρ_n) =
sum_n |c_n|^2P_n, that observables that commute with the density matrix
might have a derived Born rule following Gleason. Further, maybe operators
that do not commute then have some dual property that still upholds Born
rule. I was not able to make this work.
>>>
>>> quantum logic, and the right quantum logic is determined by the any
>>> “provability” box accompanied by consistency condition (like []p & p, []p &
>>> <>t, …).  The main difference to be expected, is that eventually we get a
>>> “quantum credibility measure”, not really a probability. It is like
>>> probability, except that credibility is between 0 and infinity (not 0 and
>>> 1).
>>>
>>> Bruno
>>>
>>>
>>> I think I ran into the issue of why Gleason's theorem does not capture the
>>> Born rule. Not all operators are commutative with the density matrix. So if
>>> you construct the diagonal of the density matrix, or its trace elements,
>>> with projector operators and off diagonal elements with left and right
>>> acting projectors (left acting hit bra vectors and right acting hit ket
>>> vectors) the problem is many operators are non-commutative. In particular
>>> the usual situation is for the Hamiltonian to have nontrivial commutation
>>> with the density matrix.
>>
>>
>> It seems to me that Gleason theorem takes this into account. It only means
>> that the probabilities does not make the same partition of the multiverse,
>> but that is not a problem for someone who use physics to see if it confirms
>> or refute the “observable” available to the universal numbers/machines in
>> arithmetic.
>>
>> Gleason's theorem applies for just one set of commuting operators,
>
>
> I am astonished by this. Are you sure you refer Gleason’s original work? I
> have seen many “simplified” proof, which sometimes add simplifying
> ```

### Re: When Did Consciousness Begin?

```
> On 1 Mar 2019, at 19:45, Philip Thrift  wrote:
>
>
>
> On Friday, March 1, 2019 at 9:08:43 AM UTC-6, Bruno Marchal wrote:
>
>> On 28 Feb 2019, at 22:47, Brent Meeker >
>> wrote:
>>
>>
>>
>> On 2/28/2019 1:17 PM, Philip Thrift wrote:
>>>
>>>
>>> The best current philosopher of (and writer about) consciousness is Galen
>>> Strawson.
>>>
>>> https://en.wikipedia.org/wiki/Galen_Strawson
>>>
>>>
>>> https://liberalarts.utexas.edu/philosophy/faculty/profile.php?id=gs24429
>>>
>>>
>>> There is a lot of his material (PDFs, articles, videos, etc.) freely
>>> available online.
>>>
>>> The main word that is synonymous with consciousness is experience.
>>
>> Which is something bacteria and plants and my thermostat have...and ability
>> to detect and react to the environment based on internal states.
>
> What the thermostat lacks, that the bacteria and plants do not lack, is
> Turing universality. That gives the mind, and even the free-will.
>
> I think free-will is just universality, and we lost it when we impose
> “security”. What makes a universal machine universal is the ability to search
> for a number which do not exist, making them able to “not stop”, and that is
> what a thermostat cannot do.
>
> Bruno
>
>
>
>>
>> Brent
>>
>
>
> Galen Strawson has an argument that makes 'free-will' something of a
> 'non-thing'. It's based on his concept of 'self'. A conscious entity (me) is
> a self in the sense that 'I am me’.

I guess this fits with the first person self, given by the “truth” variant of
provability ([]p & p) of Theatetus when translated in the arithmetical language.

> I can't really have free will since I can't choose not to be me.

We have partial control. Free-will is just the ability to decide when knowing
we lack information. It is related only to the logical self-indetermination
(not the first person indeterminacy, which gives randomness, which limits
free-will.

>
> We have 'autonomous will' but not 'free will’.

It depends on the definition that we accept for free-will. Some makes it just
absurd, like a possibility to violate physical or mathematical laws. I am not
sure that makes any sense.

Bruno

> Whenever someone talks about 'free will' not I just think of the Protestant
> denomination Free Will Baptist [
> https://en.wikipedia.org/wiki/Free_Will_Baptist] and nothing more beyond that.
>
> - pt
>
>
>
> --
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### Re: Are there real numbers that cannot be defined?

```On Tue, Mar 5, 2019 at 9:57 AM Bruno Marchal  wrote:

*> But in the “theology of the machine”* [...]

Given the fact that I don't have an infinite amount of time to read things
my rule of thumb is to stop reading whenever I encounter the T word.

John K Clark

>

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### What is the largest integer you can write in 5 seconds?

```It's easy to prove that the Busy Beaver Function grows faster than *ANY*
computable function because if there were such a faster growing function
you could use it to solve the Halting Problem. So if you're ever in a
contest to see who can name the largest integer in less than 5 seconds just
write BB(9000) and you'll probably win.

John K Clark

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### Re: Are there real numbers that cannot be defined?

```
> On 5 Mar 2019, at 15:40, John Clark  wrote:
>
> On Tue, Mar 5, 2019 at 7:40 AM Bruno Marchal  > wrote:
>
> > (And what is proof anyway?)

I did not wrote that.

>
> A proof is a construction made from a finite set of axioms using a finite set
> of rules.

Yes. That is a formal proof. But in math, proof are never formalised. Only in
logic, for the reason that the proofs are the object of study. The proofs on
that are also informal.

> If the axioms and the rules are sound

With respect to some semantics. No problem for arithmetic, as there is an
acceptable notion of “standard model”. But soundness is always defined with
respect to some informal reality, be it the standard model of numbers, or a
physical reality, etc.

> then the proof will tell you something about the nature of reality,

About reality? Reality is the model itself. It escapes all theories. But a
proof in the theory of numbers or machines, will say something about numbers
and machine. To talk about the nature of reality, you need to add some
metaphysical axioms, like Mechanism, for example. To illustrate, with
mechanism, we can say that the nature of reality is mathematical.

> if they are not sound you will be no wiser after you've completed the proof
> than you'd be after you completed a crossword puzzle. That's why it's so
> important to be super conservative when picking your axioms and rules.

Absolutely. But no theories at all can prove it is sound, nor even that it has
a model(reality). We never know that we are sound (probably assuming Indexical
Digital Mechanism (Yes-Doctor + Church's Thesis). Some logicians, like Nelson,
think that PA is already not sound, nor consistent. Few people believe him, but
each time he comes up with a proof that PA is inconsistent, the community does
its job, and usually find the mistake, that Nelson recognises quickly, but then
he continues to such the contradiction, because he is clearly convinced that PA
is already not sound. And recently, I have got that with Mechanism, PA is
unsound if taken as an ontology. We can only taken RA, that is, PA without the
induction axioms, which are already too much powerful. For the same reason, we
cannot add the axiom of infinity to RA or PA. In that case, the measure on the
computation would need the non standard computations, existing in the
non-standard model, but that has to be excluded with Mechanism, as addition and
multiplication are not computable in the non-standard models. They become
computable in a non standard sense, violating directly the Church Turing
thesis, which is normal, as the non standard natural numbers are typically
infinite objects.

Bruno

>
> John K Clark
>
>
>
>
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### Re: Are there real numbers that cannot be defined?

```
> On 5 Mar 2019, at 15:20, John Clark  wrote:
>
> On Tue, Mar 5, 2019 at 12:55 AM Russell Standish  > wrote:
>
> > The usual meaning of computable integer is that there exists a program that
> > outputs it.
>
> There is no point in arguing over the meaning of a word, but if that is what
> you mean then there is a particular form of "computation" that is as dull as
> dishwater and of no mathematical scientific or philosophical importance, in
> other words if that's what you mean then you're not wrong but you are rather
> silly.

No. It concentrates the real difficulty on the functions, and take the
primitive has trivially computable.

If that is silly, then the Church-Turing thesis is silly, the classical theory
of computability is silly, etc.

You need a good knowledge of this classical theory of computability to study
the more advanced notion of computability on the real numbers. There are many,
and there is no Church-Turing corresponding thesis.

A function (from N to N, that is what I always mean by a function) is
computable if there is a code which compute it.

What you are using here is some notion of constructively computability (we
might ask you which one). You are saying that a function is computable if we
can exhibit an algorithm for it. That means, if not only the algorithm exists,
but we can find it in a finite time, and prove it does what is requires. But
then you will lost many fundamental theorem in computer science, which
sometimes use the existence of programs which existence can be proved to be
necessarily not constructive. Some ask more: they want the concept to be
decidable, in which case we lost almost all partial recursive function. It
makes sense for security concerns, when working in a bank, but is nonsensical
in the fundamental matter where we are confronted to non stopping machine,
without us knowing if they stop or not.

If you are interested in constructive or intuitionist notion of computability,
you might read the book by Beeson, which is excellent. A good help is Dummett’s
book on intuitionism. But in the “theology of the machine”, this consists in
studying only the ([]p & p) modes of self-reference (the mathematical notion of
first person, the soul in Plotinus, the owner of consciousness, …) which
natural arithmetical interpretation is intuitionist/solipsist.

Bruno

>
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>
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### Re: Are there real numbers that cannot be defined?

```On Tue, Mar 5, 2019 at 8:03 AM Bruno Marchal  wrote:

> *The expression "Non computable numbers” appears only in intuitionist
> logic,*

If so then just by reading the title of Turing's famous 1936 paper where he
first described a device that we now call a Turing Machine you'd have to
conclude that Turing was a intuitionist, it was called "On Computable
Numbers, with an Application to the Entscheidungsproblem".

John K Clark

> On 4 Mar 2019, at 23:31, John Clark  wrote:
>
> On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal  wrote:
>
> >> I don't follow you. If the 8000th BB number is unknowable then it is
>>> certainly uncomputable
>>
>>
>> *> That is not true. All natural number n are computable. The program is
>> “output n”.*
>>
>
> I think you're being silly. You're saying if you already know that the
> answer to a problem is n then you can write a program that will "compute"
> the answer with just a "print n" command. But that's not computing that's
> just printing.
>
>
> I am even more silly. I claim that I need only to know that there is an
> answer to say that BB(n), unlike [n]BB(n), is computable, trivially and non
> interestingly, perhaps, but that follows from the classical definition of
> Turing, Church, Markov, Hebrand-Gödel, etc.
>
>
>
>
>
> Incidentally very recently Stefan O’Rear has reduced Aaronson' s 7918
> number so now we know that BB(1919) is not computable.
>
>
> Nice!!!
>
> Of course, we know only that BB(1919) = k, for k any enough big number is
>
> Perhaps tomorrow, we will know that BB(1919) = k is decidable in ZFC +
> kappa.
>
>
>
>
> So we know that:
> • BB(1)=1
> • BB(2)=6
> • BB(3)=21
> • BB(4)=107
>
> and that's all we know for sure, but we do know some lower bounds:
>
> • BB(5) ≥ 47,176,870
> • BB(6) ≥ 7.4 *10^36534
> • BB(7) >10^((10^10)^(10^10)^7)
>
> > *BB(n) is not computable means that there is no algorithm, which given
>> n, will give BB(n).*
>>
>
> Yes, so what are we arguing about?
>
>
>
> That we should not confuse the many possible notions of computable
> functions from R to R, for which there is no standard definition on which
> everyone would agree, and no corresponding Church-Turing notion, with the
> notion of computable function from N to N (or any set of finitely
> describable objects, always trivially computable).
>
> Mechanism use the Church-Turing notion. A digital brain has no real
> numbers as input; nor real numbers as output.
>
> In the classical theory of computability, a real number is seen as an
> infinite objects, and is modelled by total computable functions; or by
> recursive operator, not by the usual partial recursive functions (phi_i).
>
>
>
>
>
>
>> > *what Aaronson has shown, is that above 7918, we loss any hope to find
>> it by using the theory ZF. But may be someone will find it by using ZF +
>> kappa, which is much more powerful that ZF,*
>>
>
> It's easy to find a system of axioms more powerful than ZF, the problem is
> it may be so powerful it can even prove things that aren't true.
>
>
> That is always the risk. It cannot been avoided. Provability is a relative
> notion. No provers can prove its own consistency.
>
>
>
> Would you really trust a system that claimed to be able to solve the
> Halting Problem? I certainly wouldn't! And if you can't solve the Halting
> Problem then there is absolutely no way to calculate BB(7918) or BB(1919)
> and I wouldn't be surprised if even BB(5) is out of reach.
>
>
>
> You are right on this. The BB function computability is equivalent with
> computability with the halting oracle.
> With an oracle for BB, you can solve the halting problem and vice versa.
>
> You can’t solve the totality/partiality problem though (named TOT). Even
> with the BB oracle (or the halting oracle) you need to do an infinite task
> to solve the TOT problem. There is a transfinite set of set of numbers
> which are more and more unsolvable in that sense.
>
>
>
>
>
> > There are only 2 possibilities, a program will halt after a finite
>>> number of steps or it won’t.
>>
>>
>> > Yes. But the program which computes BB(n) always stop.
>>
>
> if it stops then it is successful but if n is 1919 then it never stops so
> BB(1919) is never computed.
>
>
> AAronson’s paper is not on the computability of BB, but of the
> undecidability of equation of the type BB(n) = k, in ZF. It is not about
> us, but about the particular Löbian machine ZF.
>
>
>
>
> >>I would maintain that you haven't solved a problem if you can't give
>>> the right answer more often than random guessing would.
>>
>>
>> *> You are restricting computability to a string notion of intutionistic
>> computability. *
>>
>
> better than you'd expect from random guessing?
>
>
> Random guessing has nothing to so with what we are talking about.
>
> My point is just that the notion of computability used is the classical
> one, and in that case, despite [n]BBn is not a computable ```

### Re: Are there real numbers that cannot be defined?

```On Tue, Mar 5, 2019 at 7:40 AM Bruno Marchal  wrote:

> *(And what is proof anyway?)*
>

A proof is a construction made from a finite set of axioms using a finite
set of rules. If the axioms and the rules are sound then the proof will
tell you something about the nature of reality, if they are not sound you
will be no wiser after you've completed the proof than you'd be after you
completed a crossword puzzle. That's why it's so important to be super
conservative when picking your axioms and rules.

John K Clark

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### Re: Are there real numbers that cannot be defined?

```On Tue, Mar 5, 2019 at 12:55 AM Russell Standish
wrote:

> *The usual meaning of computable integer is that there exists a program
> that outputs it.*

There is no point in arguing over the meaning of a word, but if that is
what you mean then there is a particular form of "computation" that is as
dull as dishwater and of no mathematical scientific or philosophical
importance, in other words if that's what you mean then you're not wrong
but you are rather silly.

John K Clark

>

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### Re: Are there real numbers that cannot be defined?

```
> On 5 Mar 2019, at 00:42, Bruce Kellett  wrote:
>
> On Tue, Mar 5, 2019 at 10:25 AM Russell Standish  > wrote:
> On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote:
> > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal  > > wrote:
> >
> >
> > >> I don't follow you. If the 8000th BB number is unknowable then
> > it is
> > certainly uncomputable
> >
> >
> > > That is not true. All natural number n are computable. The program is
> > “output n”.
> >
> >
> > I think you're being silly. You're saying if you already know that the
> > to a problem is n then you can write a program that will "compute" the
> > with just a "print n" command. But that's not computing that's just
> > printing.
>
> OK, so what about the program "print X+1", where X is the expansion of
> the number BB(8000)-1?
>
> If that's not computing something, then I'm sure I can cook up
> something more complicated to compute.
>
> I think the trouble with that, or with variations of that idea, is that they
> render the notion of 'computability' vacuous. In order to write such a
> program, or concoct such an algorithm, you need to know the answer in
> advance. That is fine, if you just want a program to compute the number 'n',
> 'n' being given in advance. But that is no help in computing a number that
> can be defined, but is not known in advance.
>
> So what people are really looking for here is a constructive notion of
> computability -- anything else has a tendency to render the notion of
> 'computability' trivial.

Then the whole recursion theory (computability theory) should be trivial, when
on the contrary it is a mine of surprising counter-intuitive results.

There is not yet clear notions of “non computable” for the constructive notion
of computability, which are dependent of the subjectivity of the
mathematicians. We got them in the talk of the “… & p” modes of self-reference.
The first person mental space is intuitionist. Indeed, it says even “no” to the
doctor, a priori.

The function defined by

If the twin conjecture is true output 0, else output 1.

Is a well defined function and it is computable, although not constructively.
It is computable, because it can be proved (easily) that its code is in the set
{K0  K1} (the two constant function [x]0 and  [x]1.

The function defined by

if phi_x(x) converges output 1, else output 0

is provably NOT computable. That illustrates that the Turing-Church’s notion of
computability is not trivial.

Bruno

>
> Bruce
>
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### Re: Are there real numbers that cannot be defined?

```
> On 4 Mar 2019, at 23:31, John Clark  wrote:
>
> On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal  > wrote:
>
> >> I don't follow you. If the 8000th BB number is unknowable then it is
> >> certainly uncomputable
>
> > That is not true. All natural number n are computable. The program is
> > “output n”.
>
> I think you're being silly. You're saying if you already know that the answer
> to a problem is n then you can write a program that will "compute" the answer
> with just a "print n" command. But that's not computing that's just printing.

I am even more silly. I claim that I need only to know that there is an answer
to say that BB(n), unlike [n]BB(n), is computable, trivially and non
interestingly, perhaps, but that follows from the classical definition of
Turing, Church, Markov, Hebrand-Gödel, etc.

>
> Incidentally very recently Stefan O’Rear has reduced Aaronson' s 7918 number
> so now we know that BB(1919) is not computable.

Nice!!!

Of course, we know only that BB(1919) = k, for k any enough big number is

Perhaps tomorrow, we will know that BB(1919) = k is decidable in ZFC + kappa.

>
> So we know that:
> • BB(1)=1
> • BB(2)=6
> • BB(3)=21
> • BB(4)=107
>
> and that's all we know for sure, but we do know some lower bounds:
>
> • BB(5) ≥ 47,176,870
> • BB(6) ≥ 7.4 *10^36534
> • BB(7) >10^((10^10)^(10^10)^7)
>
> > BB(n) is not computable means that there is no algorithm, which given n,
> > will give BB(n).
>
> Yes, so what are we arguing about?

That we should not confuse the many possible notions of computable functions
from R to R, for which there is no standard definition on which everyone would
agree, and no corresponding Church-Turing notion, with the notion of computable
function from N to N (or any set of finitely describable objects, always
trivially computable).

Mechanism use the Church-Turing notion. A digital brain has no real numbers as
input; nor real numbers as output.

In the classical theory of computability, a real number is seen as an infinite
objects, and is modelled by total computable functions; or by recursive
operator, not by the usual partial recursive functions (phi_i).

>
> > what Aaronson has shown, is that above 7918, we loss any hope to find it by
> > using the theory ZF. But may be someone will find it by using ZF + kappa,
> > which is much more powerful that ZF,
>
> It's easy to find a system of axioms more powerful than ZF, the problem is it
> may be so powerful it can even prove things that aren't true.

That is always the risk. It cannot been avoided. Provability is a relative
notion. No provers can prove its own consistency.

> Would you really trust a system that claimed to be able to solve the Halting
> Problem? I certainly wouldn't! And if you can't solve the Halting Problem
> then there is absolutely no way to calculate BB(7918) or BB(1919) and I
> wouldn't be surprised if even BB(5) is out of reach.

You are right on this. The BB function computability is equivalent with
computability with the halting oracle.
With an oracle for BB, you can solve the halting problem and vice versa.

You can’t solve the totality/partiality problem though (named TOT). Even with
the BB oracle (or the halting oracle) you need to do an infinite task to solve
the TOT problem. There is a transfinite set of set of numbers which are more
and more unsolvable in that sense.

>
> > There are only 2 possibilities, a program will halt after a finite number
> > of steps or it won’t.
>
> > Yes. But the program which computes BB(n) always stop.
>
> if it stops then it is successful but if n is 1919 then it never stops so
> BB(1919) is never computed.

AAronson’s paper is not on the computability of BB, but of the undecidability
of equation of the type BB(n) = k, in ZF. It is not about us, but about the
particular Löbian machine ZF.

>
>
> >>I would maintain that you haven't solved a problem if you can't give the
> >>right answer more often than random guessing would.
>
> > You are restricting computability to a string notion of intutionistic
> > computability.
>
> than you'd expect from random guessing?

Random guessing has nothing to so with what we are talking about.

My point is just that the notion of computability used is the classical one,
and in that case, despite [n]BBn is not a computable function, each BB(n) is
trivially, and non interestingly computable. That is why we use natural
numbers: to have a set of trivially computable primitive elements. To compute
any natural numbers, you need only to apply the successor function s a finite
number of time on 0.

The expression "Non computable numbers” appears only in intuitionist logic, or
it means non computable real numbers. But real numbers are (total) functions in
disguise, and the notion of computability on real numbers, or ```

### Re: Are there real numbers that cannot be defined?

```
> On 4 Mar 2019, at 21:34, Philip Thrift  wrote:
>
>
>
> On Monday, March 4, 2019 at 12:00:05 PM UTC-6, John Clark wrote:
>
>
> And proof is not truth.
> ...
>
> John K Clark
>
>
>
>
> Of course truth == proof in the land of radical intuitionists-constructivists.

And John does not defend such radical intuitionism. Notably by claiming that
truth is not proof.

>
> (And what is proof anyway?)

Here John did identify “computable” with some constructive notion. But that has
nothing to do with the sense of computable used in the Church-Turing thesis. It
is a different notion, and when translated in classical mathematics, you get
interesting structure, but with very restricted class of functions. Indeed, In
Brouwer’s intuitionism, or in the Topos of Hyland, it is very simple: all
functions from N to N are computable, and all functions from R to R are
continuous. That has been used by Scott to get classical set theoretical models
of Church lambda calculus (aka combinators).

Bruno

>
>
> From: Doren Zeilberger
> To: Scott Aaronson
> [ http://sites.math.rutgers.edu/~zeilberg/Opinion155.html ]
>
> ...
>
> As I have said before, there is a quick dictionary to turn all this
> undecidability babble and the obsession with related problems, like the "busy
> beaver", into purely meaningful, albeit uninteresting, statements. Every
> statement that involves quantifies over "infinite" sets, even such a
> "trivial" statement like
>
> n+1=1+n , for EVERY natural number n   ,
>
> (tacitly assuming that you have an "infinite" supply of them) is a priori
> meaningless, but many of them (including the above, and the statement that
> "for all" integers x,y,z > 0 and n > 2 , xn+ yn -zn < > 0) can be made a
> posteriori meaningful, by proving them for symbolic n (and x,y,z). So the
> right dictionary (for statements that involve quantifies over "infinite" sets)
>
> Provable : a priori meaningless (taken literally), but a posteriori
> meaningful, when interpreted correctly (for symbolic n)
>
> Undecidable: not even a posteriori meaningful, impossible to make sense of it
> symbolically
> So, like the proof that the square-root of two is irrational, Gödel and
> Turing did prove something seminal, but it was a negative result, that they
> (and you, and unfortunately so many, otherwise smart, people), in their naive
> platonism, interpret in a wrong way. So the initial "paradox" was very
> interesting, but all the subsequent "busy beaver" bells and whistles, is just
> a meaningless game.
>
> I am not saying that you are not brilliant, you sure are (and you are also a
> brilliant speaker, as I found out from your stimulating and engaging talk at
> AviFest last week), but you are wasting your talent on uninteresting
> research. Perhaps even worse than "undecidability" is your main research area
> on "quantum computing", that once again is a challenging intellectual
> mathematical game, but with empty content. The history of science and
> mathematics is full of people who had superstitious beliefs: Kepler believed
> in Astrology, Newton in Alchemy, but they did many other things besides. The
> great debunker, Gil Kalai, (who debunked the Bible Code, along with
> co-debunkers Dror Bar-Natan and Brendan McKay), has recently pointed out
> (unfortunately in his understated, gentle, way) the shortcomings of research
> in "quantum computing", and my impression is that he is right. It is indeed
> amazing how in our current "enlightened" age, that (allegedly) abhors
> superstition, such superstitious people as you (and many other, e.g. MIT
> cosmologist, Max Tegmark, another admittedly brilliant, but nevertheless
> superstitious, scientist) can be full professors at MIT.
>
> But then again, it supplies some comic relief, and some of us still enjoy
> Mythology and Theology, but it is not nice to be dismissive of people who do
>
>
> - pt
>
>
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### Re: Recommend this article, Even just for the Wheeler quote near the end

```
> On 5 Mar 2019, at 00:43, Brent Meeker  wrote:
>
>
>
> On 3/4/2019 3:54 AM, Bruno Marchal wrote:
>>
>>> On 3 Mar 2019, at 20:43, Brent Meeker >> > wrote:
>>>
>>>
>>>
>>> On 3/3/2019 4:52 AM, Philip Thrift wrote:

Here's an example David Wallace presents (as an "outlandish" possibility):
Suppose in pi (which is computable, so has a program (a spigot one, in
fact) that produces its digits. Suppose somewhere in that stream of digits
is the Standard Model Equation

(say written in LaTeX/Math but rendered here)

So what could this mean? (He sort of leaves it hanging.)

>>>
>>> Nothing.  Given a suitable mapping the SM Lagrangian can be found in any
>>> sequence of symbols.  It's just a special case of the rock that computes
>>> everything.
>>
>> Even if rock would exist in some primitive sense, which I doubt, they do not
>> compute anything, except in a trivial sense the quantum state of the rock. A
>> rock is not even a definable digital object.
>
> It's an ostensively definable object...which is much better.

Ostension is dream-able.

>
>> If someone want to convince me that a rock can compute everything, I will
>> ask them to write a complier of the combinators, say, in the rock. I will
>> ask an algorithm generating the phi_i associated to the rock.
>
> There is no particular phi_i associated to the rock.  That's the point.  The
> rock goes thru various states so there exists a mapping from that sequence of
> states to any computation with a similar number of states.

It is a mapping of states. It is like a bijection. You need something like a
morphism preserving the computability structure, which do not exist in the
rock. A computation is not just a sequence of states, it is a sequence of
states defined by the universal machine which brought those states.

There are bijections between N and Z, but only Z is a group, because those
bijections does not preserve the algebraic structure. Similarly, there is a
bijection between a computation and a movie of that computation, but it does
not preserve the causal/logical relation between the states, which is a
universal machine for the computation, and just a linear order for the
sequence, without structure, of the states.

>   Of course one may object that the actual computation is in the
> mapping...but that's because of our prejudice for increasing entropy.

OK.Now, a bijection between a physical computation and an arithmetical
computation do preserve the computability structure, that is why we can say
that the arithmetical reality/model implements genuinely the computations.

Bruno

>
> Brent
>
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### Re: When Did Consciousness Begin?

```

> On 4 Mar 2019, at 19:48, Brent Meeker  wrote:
>
>
>
> On 3/4/2019 3:45 AM, Bruno Marchal wrote:
>>  Unconsciousness is an illusion of consciousness … It should be obvious that
>> “being unconscious” cannot be a first person experience, for logical reason.
>> To die is not a personal event. That happens only to the others.
> I agree.  Except I don't suppose that all events are personal.

I don’t like this much, but physics is first person plural. Is is not purely
personal, we share the histories, thanks to entanglement, which is just the
sharing of realities in the duplication/multiplication of the observers.

Bruno

>
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>
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### Re: When Did Consciousness Begin?

```
> On 4 Mar 2019, at 19:47, Brent Meeker  wrote:
>
>
>
> On 3/4/2019 3:45 AM, Bruno Marchal wrote:
>>>   I have had two relatives die of Alzheimers and they lost their identity
>>> gradually as they lost memory.
>> They lost they memory. Not their identity, but the apprehension of their
>> identity. If not, when you ask where they are in the hospital, the nurse
>> would say “what are you talking about”. Even a corpse has an identity.
> At last you recognize the importance of the material.

Matter and physics are even more important, and more reasonable, with
Mechanism. The fact that matter becomes derivable from number makes matter
theories more well founded than any extrapolation we could do from a finite
number of observation. The material becomes unavoidable for *all* universal
machine, but its existence is phenomenological.
The quantum weirdness was to be expected, when we look at ourselves closely
around our substitution level. (Indeed exactly below that level in case we
assume us to be classically computed, it is more complicated if we have a
quantum brain, which I doubt, but that would mean our brains exploits the
massive parallelism intrinsic to the arithmetical reality.

Bruno

>
> Brent
>
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```

### Re: When Did Consciousness Begin?

```
> On 4 Mar 2019, at 19:46, Brent Meeker  wrote:
>
>
>
> On 3/4/2019 3:32 AM, Bruno Marchal wrote:
>>
>>
>> Snip
>
> A bacterium doesn't have Turing universiality, only bacteria in the
> abstract of a potentially infinite set of evolving bacteria interacting
> with their environment.  But if a consider a potentially infinite set of
> thermostats interacting with their environment of furnaces and rooms, it
> will be Turing universal too.  Turing universality is cheap.

Yes, it is cheap, like consistency, and plausibly consciousness.

But it is more cheap you might think, because even one bacteria is fully
Turing universal. The genome of Escherichia Coli can be “programmed” to
run a Turing universal set of quadruplet. Of course, the bacteria’s “tape”
is quite limited, and they can exploit their universality only by
cooperation in the long run, and so no individual bacteria can be
self-conscious or Löbian.
>>>
>>> I think that's what I said.  Except I also noted that all this requires an
>>> environment within which the bacteria can metabolize.
>>
>> That is contingent with respect of the bacteria “mental life”. All programs
>> needs a code, and an environment which run it, but it can be arithmetic.
>> Then a physical reality emerges as a means on all accessible
>> computations-continuations.
>>
>> Mentioning the environment can be misleading. If a material environment is
>> needed, matter would play some role, and there is no more reason to say
>> accept a digital, even if physical, brain.
>
> objection seems to reduce to, "But that's contrary to my theory.”

The context indicates that I was using “material” for “primitively material”.
My objection is not “that’s contrary to my theory”, but to my theorem. I am
just saying that you cannot have both “primitive matter” (or just physicalism)
and Mechanism together. Even without that theorem, the
simple-buta-pparently-not-so simple *fact* that elementary arithmetical truth
is Turing complete should make us doubt, at the least, about physicalism.

>   It's no good saying your theory is testable when you only test it within
> the assumptions you used to derive it.

Where do I do that?

>
>> In a dream, we create more clearly the environment by ourself, and that is
>> enough for being conscious, or even self-conscious, like in a lucid dream,
>> or a sophisticated virtual environment.
>
> The dream is realized by the brain and it is about elements of our real
> environment.

The human dreams are realised by the human brain, and is about element of our
human environment. To invoke “real” or “reality” or “truth” is not admissible
in science. Doubly so in metaphysics, as it begs the main question, which is
indeed about what could be real, and what could not be real.

>
>>
>>
>>
>>> So to say bacteria have Turing universality is like saying water is
>>
>> It means that with the 4 letters, you can program any partial recursive
>> function. Of course you need the decoding apparatus, but that is entirely in
>> the bacteria. It means that you can simulate any other computer, with a
>> basic set of DNA-enzyme molecular interaction. A universal machine is just a
>> number u such that for all x and y phi_u(x, y) = phi_x(y) *in principle. You
>> can implement all control structure. The operon illustrates a
>> “if-then-else”, and the regulation apparatus is enough to get universality.
>> René Thomas, in Brussels, has succeeded to make a loop, with a plasmid
>> (little circular gene) entering in the bacterium, and then going out,
>> repetitively. It is even a “fuzzy computer”. Some product are regulated in a
>> continuum, depending on the concentration of the metabolites. When I was
>> young, I have made e project for a massively parallel computers which was a
>> solution of bacteria (E. coli) and bacteriophage. One drop of it could
>> process billions of instructions in a second. But the read and write was
>> demanding highly sophisticated molecular biology. I think that such ideas
>> have more success today. After all, molecular biology studies “natural
>> nanotechnology”.
>
> You're wrong.  The environment is essential.  The fact that DNA can encode
> functions means nothing without the ability to read and execute the code.

You can easily program a decoder of instruction, and a decoder of addresses in
the arithmetical language, and *all* models of arithmetic implements them.

> RNA, proteins, krebs cycle, and proton pumps are all necessary for that.

That is carbon chauvinism, with all my respect. I am a lover of Krebs cycle,
and even more Calvin cycle (in photosynthesis). My initial inspiration of
Mechanism came from Molecular biology. But nothing there has been shown to be
non-Turing emulable. ```