ThinkArt Lab Animation: A.T. Kelemen© November 12, 1998 Dr. Rudolf Kaehr

## 1 A fundamental theory of the natural

If there is anything left in this world we live which is still untouched and natural then it is the naturalness of the natural numbers-and nothing else.

Brian Rotman"´Natural´ because they are given at the outset, taken for granted as a founding, unanalizable intuition, outside any critique that might demand an account of how they come or came-potentially or actually-to ´be´."0.5 Le nombre regle les representations culturelles."0.6. Le nombre, evidement, regle l´economie, et sans doute est-ce la ce que Luis Althusser aurait appele la determination en derniere instance" de sa suprematie. L´ideologie des societes parlemantaire moderne, s´il y a une, n´est pas l´humanisme, le Droit du Sujet. C´est le nombre, le comptable, la comptabilite."Alain Badiou, Le Nombre et les nombres. Seuil 1990And why not Leopold Kronecker?"God made the integers, all the rest is the work of Man."As Natural as 0,1,2Philip Wadler. Evans and Sutherland Distinguished Lecture, University of Utah, 20 November 2002."Whether a visitor comes from another place, another planet, or another plane of being we can be sure that he, she, or it will count just as we do: though their symbols vary, the numbers areuniversal. The history of logic and computing suggests a programming language that is equallynatural. The language, called lambda calculus, is in exact correspondence with a formulation of the laws of reason, callednatural deduction. Lambda calculus and natural deduction were devised, independently of each other, around 1930, just before the development of the first stored program computer. Yet the correspondence between them was not recognized until decades later, and not published until 1980. Today, languages based on lambda calculus have a few thousand users. Tomorrow, reliable use of the Internet may depend on languages with logical foundations. "http://homepages.inf.ed.ac.uk/wadler/topics/history.html#drdobbsOur whole economy of living and thinking would immediately crash, if the slightest change in the nature of natural numbers would occur.

If someone wants to achieve to be nominated as the most cranky mind, the crackpot par excellence, he/she/it should try to prove a paradox or even a defect in the very nature of the natural numbers.

Also such efforts are not unknown, they didn't have any impact on the nature of our natural numbers. It seems to be much more accepted to invent new and deviant logic systems than to change anything in arithmetics.

In my study, I will not touch these tabus. In contrary, I will accept them in all their principality, I even will celebrate them in disseminating them in their whole sacrality.

In doing so, the exclusive nature of the natural numbers will boil down to a very mundane activity in our cultural, that is, artificial world.

The naturality of the natural number system, as we know it, will be entangled in an activity of increasing artificiality of multitudes of natural number systems.

Also there is no culture without numbers, numbers are not cultural, but natural. They are the very nature in/of our culture. To transform this situation will change radically what we will understand by culture. The most advanced development of this classical arithmetical trance of naturality is still the global movement of digitalism and its technology.

In other words, my old question is still virulent:

(ISEA ´98)What´s after digitalism?## 1.1 Natural number series

## 1.1.1 Natural numbers as models of fundamental abstract systems

"A first attempt at a theory to describe numbers begins with a fundamental abstract type called nat0 as follows:

Any theory that consists of a signature without any equations is said to be fundamental because it generates all possible strings of symbols without defining any equivalences between the strings. In this particular case the signature contains an arity-zero operation called

and an arity-one operation calledzero. These operations generate the following infinite series of expressions:suc

, ...zero, suc(zero), suc(suc(zero)), suc(suc(suc(zero)))in their Herbrand universe of the type.

The only

applications of these operators are the constantwell-formeditself or succesive applications of thezerofunction beginning withsuc.zeroSince there are no equations in this theory, every element is

and we obtain andistinctnumber of one-element equivalence classes.infiniteOne very obvious

for the possible elements of theinterpretationnat0 is the series of denary numbers {1, 2, 3, ...}, settingabstract typeequal to 0,zeroequal to 1, and so forth, ..." Michael Downward, Logic, p. 181suc(zero)This is well known, well established and usefull and for some strange reasons it is called

. And it offers a stable fundament for the natural number series and all other types of linearly ordered series, too. At least there are enough people who strongly believe in that.word algebraAs we see, and will see in the following, natural numbers, despite on being natural, are not naturally accessible in mathematics. They need all sorts of sophisticated notational systems and interpreting mechanisms.

Technically, the natural numbers are accessible and representable only

and not to some real-world concrete inscriptions. Insofar, there is a structural gap between our intuition of natural numbers and the formalization of this intuition.up to isomorphism"Because most of the stuff about natural numbers and their formalization is standard I will make extensive use of citations of this material. The way a bring these citations together will give us new insights about the different strategies of formalizing the intuition of natural numbers. Mostly we are reading one of these ways of presentation and not much comparison is done. My approach is not only confronting the different approaches but also putting them together in a new light of mutual contrast and explanations.

The

emphasis the aspect of step-wise construction by rules applied to an initial object. This shows us more the internal structure of the type of construction.Stroke Calculus approachThe

approach develops an understanding of natural numbers out of a special set theoretical operation, bracket-operation for sets, based on a logical definition of the empty set which in itself is not very self-evident.Set TheoryIn contrast, the

approach emphasis on the external relationships of the constructors and gives us an explication of the intuition of natural numbers up to isomorphism.Category TheoryAll these approaches make it quite clear that the naturality of natural numbers is not as easy captured as it is suggest to be for a natural intuition of a natural object.

Are the" intuitive natural numbers categorical? That is, is the description of natural number as clear and definitive as we usually take it to be?

This was no idle question for Frege who in the Foundation of Arithmetics attempted to achieve an absolute and clear description of the natural numbers. Any denial of categoricity has importand consequences.

Isle p. 111Whenever we define a class of mathematical objects via inductive definition and the proceed to establish results about objects in that class we make tacit use of properties of certain functions."What is the importance of the natural numbers?

As the citations show, natural numbers are of importance on the very base of our culture and technology.

Natural numbers deliver the prototype of any constructivist theory, even of any theory of construction.

How are natural numbers notated?

Numbers, numerals, marks, ciphers

What is natural to natural numbers?

What would be unnatural for natural numbers?

gaps, multitudes, obstacles, neighbors

## 1.1.2 Natural numbers in a constructivist Stroke Calculus

In a stroke calculus the representations of natural numbers are produced by three rules.

R2) Given a set of strokes (call it X) write down X1.

R3) Now apply R1 once and then apply R2 again and again.

"An understanding of the "structure of the natural numbers" thus consists in an understanding of these rules. But what has actually been presented here? Rules R1 and R2 are fairly unambiguous, in fact, one could easily use them to write down a few numerals.

But rule R3 is in a different category. It does not determine a unique method of proceeding because that determination is contained in the words "apply R2 again and again".

Isle, p. 133But these words make use of the very conception of natural number and indefinite repetition whose explanation is being attempted: in other words, this description is circular."Even if we accept this criticism of the rules, we have to accept, that rule R2 demands some preconditions, at least, we have to add the new stroke in line with the other strokes, and not somewhere else, e.g. behind the blackboard. If we use the quite misleading terminology of sets in rule R2, the new stroke has to be written in the domain of the set and not outside of it.

But why should we accept that, if it is not explicitly asked? Therefore, the game is not so clear as it should be. The presupositions of the stroke calculus is

linearityof repetition andatomizityof its strokes, short: semiotic identity.All these presupositions may not be very natural, they are not pre-given, we simply have to learn them, that is, to internalize them by education.

We may interpret the stroke calculus as an example which starts the numerals with the initial object 1and has two rules as constructors to construct the object, that is any natural number.

## 1.1.3 Dialogical foundation of the natural numbers

This constructive approach of the

can be made much more explicit and more adequate in formalizing the intuition of natural numbers in the framework of the dialogical approach. Especially the 3 rule has a more advanced treatment in the dialogical setting explained by Lorenzen.Stroke CalculusThis type of construction is more a type of reconstruction then a construction ab ovo of the natural numbers. With this distinction we have a more explicit idea of the process of formalizing an intuitive idea of the natural numbers because the intuitive knowledge of natural numbers is directly confronted with the formalism.

To ask if 10

10is a number means two things, first I have an intuitive knowledge about 1010to be a number and second, I have a formalism to answer the question in applying rule 3 as long as I need to construct 1010or to get an agreement with my proponent in the dialog about the construction of the intuitive number 1010. Intuitive means in this case that I have a notational system to write my supposed number but I don´t have a procedure to produce this number, therefore it is the task of the proponent to use his formalism.This setting also tries to escape the circularity of the situation, to ask if 10

10is a numbers presuppose that this object is a number. To ask if 1010is a number means to ask for a procedure to produce step-wise without violating the intuition of counting the desired number 1010.To put this situation into a dialog between opponent and proponent seems to be an explication as an interlocking game between intuition as pre-understanding and formalism as construction which escapes the purely formalist thematization which results in circularity.

But all that doesn't mean that the game between intuition and formalisms has stopped and we are now in possession of an ultimate formalism which corresponds to the very intuition of natural numbers. The results are limitations, as we can learn from Kurt Gödel, there is no strict formalism which would be able to formalize in full the idea of natural numbers, what means, the idea of infinite induction which is postulated in the "and so on" of rule 3 of the stroke calculus.

Diagramm 1

Lorenzen´s dialogs## 1.1.4 Semiotics of natural numbers

Max Bense, Zahlzeichen - Zeichenzahl

## 1.1.5 Natural numbers in set theory

We may interpret the set theoretical definition of the numbers as an example which starts the numerals with the initial object nil, the

. That is, with a negation and with non-existence (in set theory). Thus, the empty set is defined internally, and not from an external point of view, by its attribute to be the set of all objects which are not identical with themselves. The presupposition clearly is, that there is no such object in the universe.empty setAgain, there is no escape, circularity is at the very beginning.

## 1.1.6 Natural numbers in Category Theory

We may interpret the category theoretical definition of the numbers as an example which starts the numerals with an initial object without giving any information of its internal structure, but only about its external relations to other objects.

Diagramm 2

Kategorientheoretisches Diagramm NNA natural number object consists of an object and two morphisms

such that for all objects A and all morphisms a: 1 --> A, h: A --> A there exist a unique morphism f: N --> A making commute the diagramm NN.

Diagramm 3

Short-Diagramm NNAn important fact is that any two terminal objects (as well as any two initial objects) in a category are uniquely isomorphic. In other words, if T and T` are two terminal objects, then there is a unique isomorphism between the two. Because of this, it is customary, to collapse all terminal objects into a representative and talk about the terminal object.

As we see, 1 and 1* are isomorphic in respect to f and f*.

The categorical approach to characterize objects and morphisms in terms of their relation to other objects and morphisms has the particular consequence that universal properties specify objects onlyup to isomorphism".Definition: Objects A and B are isomorphic if there exists morphisms f: A --> B, f*: B --> A such that f*.f=iA and f.f*=iB

## 1.1.7 Natural numbers as numerals in an arithmetical game

## 1.1.8 Natural numbers as visitors of different modi of thematizations

## 1.2 Natural numbers and computability

"We offer this conclusion at the present moment a a. And to our mind such is Church´s identification of effective calculability and recursiveness.working hypothesisThe success to the above program would, for us, change this hypothesis not so much to a definition or an axiom but to a.natural lawOnly so, it seems to the writer, can Gödel´s theorem concerning the incompleteness of symbolic logics of a certain general type and Church´s result on the recursive unsolvability of certain problems be transformed into conclusions concerning all symbolic logics and all methods of solvability.""Actually the work already done by Church and others carries this identification considerably beyond working hypothesis stage. But to mask this identification under a definition hides the fact that a fundamental discovery in the limitations of the mathematicizing power of Homo Sapiens has been made and blinds us to the need of its continual verification." Emil Post, 1936## 2 Cloning naturality

Today it seems that there is no reason to not to clone and replicate the naturality of the natural numbers with their ultimate Herbrand universe.

What is pre-given and natural, should be replicated to loose its magics.

(We only can understand what we can reproduce. AI)

To be modest I start with the replication of the Herbrand universe into only 3 clones. Each clone has its own word algebra with all its distinctions. But the distinction between the original system and its clones disappears in the sameness of the systems. These clones are not models of the abstract system, they are abstract in themselves. Insofar they are all the same, and we have lost the original system with which we started. That is, the original is only original for its role as a starting point of the process of replication. As products all these systems are the same.

The metaphors

andclonewill help us to surpass the dictatorship of identity on all levels of our thinking and writing. The idea ofcloningas logically different from identity and diversity and also not rooted in them, will lead our thoughts of distributing and mediating systems to realize a construction of cloning the naturality of the natural numbers. The poly-contextural approach to the new category of sameness, with its ontological, logical, semiotical and arithmetical consequences, goes far beyond such concepts like multi-sets" which are commonly used for describing replication in biological systems.samenessThe construct nat0

(3)denotes the 3-fold replication of the abstract type nat0. Because these clones of the Herbrand universe are living together we call there spared space ultra-Herbrand multi-verse or simply theirmulti-verse. Also the replicands are all the same but not identical they can be distinguished and are therefore countable, and here we have 3. For separating the replicas and for bringing them together in their multi-verse I introduce the operation of dissemination DISS. In other words, the operator of dissemination DISS produces a distribution and mediation of the systems under consideration.The complementary movement of the process of cloning the naturality of natural numbers is given by the idea, that our natural numbers are not so natural as we have learned to think or believe. It is equally reasonable to think that the unicity (uniqueness) of our natural numbers is the result of a powerful squashing and squeezing together the multi-linearity of numbers by force to the uni-linearity as we know it. Therefore we have a chance for a desedimentation and deliberation of the numbers from the terrorism of linearity to a free play of writing opening up not only a multi-linearity of numbers but a living tissue".

This idea is easily supported by Aristotle´s condemnation and fight against Platonist and Pythagorean ideas of numbers.

But also by the historical movement of creating a monetary equivalence between different money stems.

## 2.1 The conceptual graph of the abstract object nat0

To explain, in a first step, the concept of distribution and mediation, i.e. the concept of dissemination of formal systems, I introduce the notion of a

. A conceptual graph shows the conceptual dependency structure, e.g. the dependency structure of the notions of a system or an abstract object.conceptual graphIt will offer us a practical tool to construct the chiastic structure of disseminated objects in general.

Diagramm 4

graph of nat0The arrows in this diagram represents conceptual dependencies in the notion of nat0. The notation

the concept of opns varies as the concept of sorts varies.

In particular, it means that the concept of opns, the one that we have in mind, cannot be independent of the concept of sorts and neither can a particular opn be independent of its particular sort.

means that the concept of sorts varies as the concept of nat0 varies.

Therefore the notion of opns varies as the notion of nat0 varies:

In a conceptual diagram, 1 represents the absolute. The notion

expresses that the nat0 notion is absolute, for it tells us that the nat0 notion varies as the absolute varies - which is not at all.

Normally the notion of the absolute is not included in the definition of an abstract object like nat0, simply because we presuppose that there is anyway one and only one such concept of an abstract object. But for the purpose of disseminating abstract objects it is exactly this part of the definition of abstract objects which has to be deconstructed, i.e to be distributed and mediated. Abstract objects have not to be confused with the multitude of models of abstract objects as concretizations of the abstraction into the concrete world of formal and not formal objects and entities.

In other words an abstract object, in this sense, is an institution.

## 3 Unicity, Intuition and Explication

## 3.1 Aspects of the interplay between intuition and formalism

Mostowski, Thirty Years of Foundational Studies, 1966, p. 33"Hower much we would like to ´mathematize´the definition of computability, we can never get completely rid of the semantic aspect of this concept. The process of computation is a linguistic notion (presupposing that our notion of language is sufficiently general); what we have to do is to delimit a class of those functions (considered as abstract mathematical objects) for whichexists a corresponding linguistic object (a process of computation)."

(GoguenTruth is invariant under change of notation."

Goguen"I think it is fair to say that most mathematicians no longer believe in the heroic ideal of a single generally accepted foundations for mathematics, and that many no longer believe in the possibility of finding "unshakable certainities" upon which to found all of mathematics."

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