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The bit, atomic but clever

Part of this text has been transferred and edited in diccan's bit article.

1. Digital and binary

The definition, and crucial quality, of the digital is that it can be reduced to bits and sets of bits. So, it can be seen as a set of binary values. For instance as 0s and 1s along the numeric scale of integers (N). Or a true/false boolean. Or a positive/negative image...

But this reduction to bits must not take to see this world as a raster bitmap image. Its most important structures operate at a higher level. Languages have words and syntactic laws. Physics groups atoms ino molecules. Genetics group pairs by 3 (then 6 bits) into codons. These may reduced to bit strings by coding and decoding rules, by parsers and compilers. The rasterization is only the last resource when a digital universe needs to communicate with or through matter.

Then, non digital beings (continuous, analogue...) may be represented or generated along two ways:
- digitization/rasterization, for instance the pixels of an image
- naming of beings and formal descriptions of their features and relations.

A digit is an element of number representations (or command devices) , from 0 to 9 for decimal numeration, 0 to 1 for binary, 0 to F for hexadecimal. The binary digits reach a wider range, since they may used to represent not only numbers, but also any two-valued feature. And they are conveniently implemented physically with logical circuits. However, they are not practical for the human cognitive system in general. Machines use bits by sets of 8 bits or more. Nature uses pairs of bits, grouped in genes, or sequences of (binary) pulses in the neurons.

The streaming of text into bits, and the reciprocal destreaming are done without loss, as long as :
- text conforms to the syntax of the used language,
- non syntactic expressive complements do not matter (such as font style and page setting for written text, and prosody for oral text). .

We shall say more about bits and matter in the "Incarnation" chapter, but we should never forget that, in nature as in artefacts, digital beings never operate in total independence of matter. In the reproduction of living beings, the combination of genomes will give a new being only through an epigenetic process which is not totally digital.

2. The elusive nature of the bit

What is a bit ? The answer is not easy since, for example :
- a bit is always material as well as non-material; but it is not necessarily a definite location on a material substract, it can also be a impulse along a pair of wires, a fiber optic or a neuronal axon; the nature of this matter may be important, but remains secondary Vs. the meaning

- a bit is always a signifier and a signified ; the problem of meaning vs. digital has been seen at least from Weaver and Shannon book ; the more you digitize and the more the meaning goes out of the parts into their disposition, the structure arount the bits ;

- an isolated bit has no meaning ; it acquires meaning throuth its material environment (the neighbouring bis) or through its "semantic" environment, typically, a pointer on a boolean from an object ;

- a bit is never reached directly ; the lenght (or weight, cost...) of the way to access it could be a quantification of the "degree of digitization"; for example, a procedural texture is more digital than a sampled one since with the former one, the access path is coded, when, in a sampled one, the access is a map ; bits cannot be read with the naked eye;

- this digital degree could be interesting in languages ; it is not by chancde that the occidental alphabets have used around 25 letters since the Phoenicians;

- more generally, there is a "granularity", partly necessary by human (animals ?) cognitive structures. See for example the introduction to Bourbaki.

A mathematical expression of the digital universe is {0,1}*, the infinite set of 0,1 pairs.

Digital cosmology gets a good trump in its base, the bit: solid, robust and perfectly atomic in the etymologic sense of unbreakabilty. There is no thinner nor smaller logic element than the bit, no smaller information unit, an no smaller element in decision and action.

Some of these points may nevertheless be discussed.

- a ternary logic would, for some experts, be better, since the mathematical constant e is nearer to 3 than 2;
- in some electronic devices, or parts of devices, a "three states logic", the third state being somehow neutral, is more efficient;
- modal logics have attractive aspects that not always reduce to binary;
- a bit is not the least possible information quantity ; its proper value is reached only when the two positions in the alternative are equally probable ; if one out of the positions is more probable than the other, the informational quantity decreases, and vanishes when one position is certain.

Some other considerations :
- is a bit is a particle, and not an atom ?
- a bit is a kind of balance, a dissipative structure (a priori in a digital universe, and on full right in the physical universe);
- bits may be seen as electrons running through logical gates, which are stable structures of nuclei which determine the flows; hence, immediately, the irreversibility of logical functions.

3. One bit, two bits...

A bit alone has neither meaning nor value by itself; it is not even perceptible. at limit, a sole bit does not even give the direction of transfer, implied only by the transmitting device

Nevertheless, the mere existence of a bit brings up an elementary optimism. If there is a bit, that implies that, somehow and somewhere, order is greater than disorder, structure more powerful than chaos. At least locally. From the most elementary level, we can then subsume the presence of a system funding values, yields, operational capacity, and at least a difference versus indifferenciation.

But that, by itself, places the bit inside of a system which lets appreciate it. We shall deal later with an hypothetical "general system". Let us begin with small bit configurations, which give consistence to their constituents.

4. Discrete and continuous

Could this model, and under which conditions, build the continuous ?

Discrete beings may be processed continuously, in a large measure. For instance, cookies bore by a rolling chain during the baking phases. Reciprocally, continuous products may be "discretized" for processing (batch processing in chemical industry or information systems).

The term "transaction" says, generally speaking, that a batch of operations is applied to a batch of something, often by an exchange between two processors. This term is used mainly for money and information. Formal atomicity rules have been expressed for a correct and fail-safe transaction processing (double commit, Acid rules).

Continuous and discrete are never totally separated. "Every morphology is characterized by some qualitative discontinuities in the substrate" René Thom. Mathématiques de la morphogenèse, [Thom, 1972]. In other words, digital is always present, even in analogical systems (dissymmetry?)

The real line R

From digital beings, we can build the real line R, to find again the continuous. This construction cannot be done "directly", since it supposes infinite somewhere. But a way through formal, the infinite potentiality of a "meta" level in regard to the inferior level. Projective spaces ? Hence, one could finish up on analogue computation, if there is still something to work out of it (not totally, see Bailly-Longo).The cut in R real line construction is dual of the fundamental N cut. It is their combination which gives this impression of "total" recuperation of the real.

Objective: given a finite set of points (e.g. in R2), let us look  for the simplest function offering a sufficient account (or generation) of the points. The optimum is computed on the equation length over the precision. This gives another type of complexity, "with loss", from a strictly logical standpoint. An even possibly with gain, if the original bitmap is considered as a forced digitization of underlying continuous beings (e.g. circles), a sort of platonic cave.

Continuous beings, and matter itself, emerge only by irreducibility of logic forms (square root). Matter (or animality) is what supports keeping on, bridge between, waiting for (let us hope) a more subtle formalism which re-establish continuity, or even better logical coherence.

Physically, continuity is an illusion (quanta), or better a construct, no more "natural" than digits.. Continuous functions are only approximations to express the large numbers. It is built by the brain. And is a sort of necessity to think as well as to act.

A continuous function is continuous in respect of the being it describes, but nevertheless is discrete in its expression (text). It draws a bridge, symmetrically to physical matter, thought as an infinite base for bits and quanta.

Contiunuous is a vue de l'esprit. Afforded by large numbers, and stong simplificatins.
Fine fragmentation. Power. a sort of liquidity. Emulsions, pastes.

In gases and liquids, light rays (sound) propagae. Isotropi

5. Digital and analog/analogue

The distinction applies to representations (here analogue is more frequent) as well as to devices and specially electronic circuits (here analog is more frequent).  At first sight, the distinction seems perfectly sharp and clear. In practice the border may be fuzzy.
This opposition is not orthogonal with the discrete/continous opposition.

5.1. Representations

See also representations.

An analog representation
- is continuous (if the represented entity is continuous) and
- is obtained from the original being with some kind of continuous function: optical transfer, electrical value transformation, etc.

Digital representations cut everything in parts, normally in bits. Digital devices and computers use logic circuits.

Typically, the difference is illustrated on a car dashboard, with needles on dials (analogue) opposed to digits (here, numerical digits)

In most cases, the border is not so sharp.
- digital representations, at high resolution, appear as continuous and analog
- for ergonomic or aesthetic reasons, digital data are frequently presented in analogue form
- quanta theory

The difference between analog and digital lies in the mapping of being structures into bits. Analogue mappings transfer, more or less, the spatial relations of the beings into the arithmetical succession of bits. Digital mappings use the arithmetical succession as a support to any kind of relation internal of the beings. That is the basic addressing system

Example: on a random sequence of 0s and 1s (not too compressed, let us say one among ten), density computation at one point, more or less thin, we get a continuous function using interpolation or smoothing.

Kinds of analogies
- myth
- anthropomorphism, and in robot design
- logical inference as analogue of a process
- iteration as an image of the wheel.

A digital representation is built by three basic cuts:
- the representation is not the represented being (the map is not the territory)
- cutting the representation into bits
- the representation structure is not (or at least, is not necessarily) homothetic to the represented being. (Analogue/digital). This last cut is what makes the difference between analogue and digital. (better words should be found).

and : cutting the represented into bits ? and the correspondenc between the cuts.

A new bit is cut, a new being is borne. At some moment, the causal vice loosens. A new complexity level is reached, with potential DR for undetermination and recursivity, letting appear a new autonomous being.

5.2. Analog devices

Analog circuits. See definitions and catalog of circuits on the web.

An analog computer (that existed until the 70's) uses such functions.

About the Chevalley/ Zarisky dialogue (algebra vs. geometry) :

- they start from a word , then “I mean”
- it is not directly ontological
- after that, the real issue is : what to do with that, what can ve modify, what gameplay do we have
- with geometry : displace, change the form (topology), dip it in an 3D space
- with algebra : write the formula, a formula for f, and any thinkable formula.
Let us note that Chevalley has chosen a 2D space

Hence, we could ask : what will to the computer specialist with this « I mean »? "Bézier", for instance? or the neurologist ?

5.3. Any object is “hybrid”

Thesis. No being is totally digital, nor totally analog. There remains always some analogies in the coding, the code structure, etc. At least in the order of bits. Even a non physical representation is not totally analog.

6. Concrete/abstract

When abstract beings tend to grow and increase their "resolution", concreteness seems to emerge by this sole fact. Abstract is seen generally (if not always) as shorter than concrete. Anyway, the "final" concrete is matter, and is supposed to give an infinite potentiality of drilling to more and more information.

There is a kind or symmetry:
- when abstract grows, concrete emerges out of it ;
- when concrete knowledge (measure) grows, abstract must become more finely cut, then more massive.

General rule (relation between extension and comprehension) : the more we know about a being, the more bits contains its description, and the less many are beings answering such a description. But this rule is not always true in the concrete. Then it must be taken with moods.

Absolute concrete may be taken as a sort of limit, of what is accessible by all beings in DU (and possibly without conventions ? )

An abstract system is made of concepts, totally defined by hypotheses and axioms of its creators ([ Le Moigne 1974] after Ackoff) . Is it not confuse abstraction and logics .?
Abstract beings have a limited and definite number of psychemes. (Lussato, unpublished1972)

This issue is partly quantitative. It remains to qualify the gap (here, "excess" of code) in order to access to an individual, and to describe it. For a population, the minimum is the logarithm of the population. For the French social NIR (security social code), 1 38 08 75 115 323, that is 1013, of 6.107 inhabitants. Whence the overhead ?

Measure abstraction. If the message is short, an external context is necessary to understand it. Then:
- either this context is itself concrete,
- or it is abstract also, but there is, comparing to the longer number, the concrete aspect of the link.
Bouasse: on the corporate correspondence sheets : Telephone
The 67 at Lasseube (France)
19/33 16/47 32 18 67
00 05 47 32 18 67

Thesis : A hyper code without redundancy entails hiearchization loss. Ant a wrong bit spoils the whole system, if there are non longer stronger and weaker beings.

First, on a given object, unique, for instance a person, the number or descripted beings no longer changes as soon as it has been identified by a sufficient number of bits (population number logarithm). Then we can tell his history, etc.

Similarly, when we learn more and more about a being type. That does not augment the population cardinal number.

Then the rules apply only partially.

The time comes when abstract connects with concrete : segment one (from the "one to one" marketing language), "the one who...". du "celui qui". Evolution of information quantity about a consumer. At start, the customer does not even exist in the information system. It records only anonymous (or difficult to group, as with insurance contracts) operations. Then there is growth.

Abstract phase: set of beings having no intrinsic meaning, but representing (or coupled with) concrete beings, being classes or psychological beings. These beings (symbols and signs) are transported by a concrete being, in physical phase [Lussato, 1971].

Possibly : a concrete object is unique as an instanciation.

7. Natural and artificial

Artificial beings:
- are synthesized by living entities (though generally not with a clear anticipating view); or by other automata?
- can mimic natural beings, but lack some or other aspects of the real natural being;
- may described in terms of functions, aims, adaptation;
- are often considered, in particular at design time, in imperative terms more than descriptive (HA Simon, 1969, from the French text).

The natural beings are the non artificial ones. a being may be said natural if one does not know any digital being that has made it "consciously". That is generally easy to say of non living beings found in "nature", as stones. Living beings can be considered as natural since the origin of life is unknown, as far as the biological reproduction is compulsive and not really controlled. On the other hand, we shall call "artificial" beings that result out of a "conscious" activity, such as bird nests or beaver dams, and of course the products of human activity. This distinction is problematic, as well as the word "conscience" itself, which is generally admitted as a mainly human feature, somehow shared with animals, but not at all with machines. Does technological progress leads us to conscious machines and to a merging of natural and artificial world (or, it may be also said, as a complete "artificialization" of the world) ? That is an old issue, which emerged mainly in modern times, but has very ancient historical roots.

As long as life is so different from non-living existence, we may generalize "artificial" to all artefacts due to living beings, at least if they are external to their bodies, relate to a sort of "conscious" activity.

So, we call natural all beings not explicitly made by living entities, including these entities themselves, as long as they cannot reproduce themselves by explicit processes.

However, artificial entities can reproduce themselves explicitly, at present only under the elementary form of virus. Then virus and their productions may be considered as artificial. If some day, and that day will perhaps come soon (or perhaps never), mankind succeeds in creating life from non living beings, the border will lose its present radically impassable nature. On the long term, men would be able to make living beings, and robots would be able to make human beings...

The artificial world defines itself precisely at this interface between internal and external environments. It shows us how to reach aims by adaptation of the first to the second ones. HA Simon 1969

The world where we live today is much more an artificial world, man made, than a natural world. A ploughed field is no more a natural element than a tarred street. A Simon, 1969

Idea : The global digital universe (if it exists) is natural. A bit is natural or not.

8. Pixelization and vectorization, the two ways of digitalization

See the digitalization chapter.

9. Matter/Form

Any real entity is
- partly handlable, named within a formal system. I call that form (even if it is rather different of the traditional decensions or this term), abbreviated in F. Form is textual, then can be fragmented down to bits. A leas "materially"... but meaning ? That can also been called hardware, but with nuances. Any "hardware", digital at least, has a formal part.

It would be better to say: we interact with the external world by two different ways. Formal and material.
But it is truly also for other objects than us (DNA).

any bit has to be materially located/operated/transmitted.

- partly out of reach for logic. down to quanta. I call it matter, abbrevited in M. F does not exhaust matter.

Matter is partly infinite. At least downh to quanta, I can constantly get new bits about it. And "materially", it is always larger than the formalized part of it
Matter is partly finie. Even the universe globally has a mass.

Matter is not only solid, but also liquid, gazeous, rays/fields, locality and frequencies
Form cannot be reducet to rational description by us. Bits are present in DNA.

Form is partly infinite : recurrence, recursion. Measure the level of recursivit
Form is partly finite: number of bits in any sysem. Storage volume. Change speed limits.

There are relations between the finitudes. Moore's law.

There are relations between the infinitudes ?

There are realtions between formal inf. and matter fin : limits of computation power, delays, storage capacities Grain, quanta.
between formal fin. and matter inf?

I exclude "spirit", that is form without matter or not formally describable (ineffable).

The form system, may be considered a space, basically {0,1}**NN (NN is the number of bits).
The transitions from one point to another one form a graph. Logical graph..
How is recursion represented in a graph.

For pracical (but deef) reasons, the number dimensions of that space is (strongly) reduced by construction rules.
Proofs are a kind of graphs.
Decisions create constraintes in a family of grahs. Random or not.

Then there is also the actual capacity of moving in the graph. Computation power, network.

Formal "imperfect processes" and moves in the graph can be used

Some moves in the F graph are determined by its material base. can be used to accelerate the process and lodge it in its practical possibility.
In particular the clock and its frequency.

We can control/influence the entity by both material and formal systems (hardware mechanics, soldering Vs. software).
With intermediate solutions (firmware).

Both sides are respoectable, and can be used. e.g. random throug mechanical imperfections (Meta of Tinguely) or CERN modules, or sortware pseudo-random. Using also semi random, with clock and

Analog is within a formal system. A copy matter strctures.

Matter/form exchanges by sensors/actuators.

The cooperation M/F is quite dialectical, works by opposition. Matter is not happy to constrainged by the formal grids. Form is not happpy to be limited, to be constantly at risk of being dissolvec or last.

There is a constant competition in matter (forces oppositions)
in forms. patterns fight. Meme

Possibly a double way of bits into matter :
- a small surface into a raster (e.g. pixels, voxels)
- a partition of a space into two parts (without material border)

Exchanges are always analog/material at some level.
the OSI layers

10. Quantity and infinity

Bits and infinity : through recursion.

Digital/analog as a relationship

The emission levels :
- not properly an emission, juste an evolution of the emitter or environment
- voluntary, finalized analog, gesture
- voluntary, coded; the emitter has an D/A generator.

The reaction is analog if the recevir reaction is continuous. Generally simple
- the simplest is no evolution at all
- proportionality
- exponential
- feedback

reflexion/refraction of light

The reaction is digital
- no evolution at all
there is a threshold. - one bit evolution (but possibly with complex consequences, typically atack/flee
there is a set of patterns, with a feature space and an algorithm to select the best answer

Life. DNA as emitter for embryogenesis.

Or course, the "conscience" problem

Then coding properly. education of a doc
Conventional set of sigbnals (road, marine) Protocols.
An external reference to the communication. Language (common memory) then monuments, books... Internet. Both memory and net.

Binary sex male/female. Could not be ternary ?
pure information has a null damping factor.


  environment, or emitter analog digital
  temperature change follows state change, attraction/repulsion
  color copy, colored adapted behavior
  stimulus   response

9. A global universe and its parts

See the global world chapter.

10. Varia to be developped and edited

Clever and selfish ... These words are anthropomorphic. But, as with Dawkins it may be an efficient scheme, a useful “theoretical fiction”.