Theory of named sets (Burgin)

In: Structures in Mathematical Theories, San Sebastian, 1990, pp. 417-420

Burgin M.S.

THEORY OF NAMED SETS AS A FOUNDATIONAL BASIS FOR MATHEMATICS

The most recognized basis of modern mathematics is set theory. But at the same time other mathematical constructions appeared (some of them recently and others many years ago) that in some sense are more general than sets. These are fuzzy sets, multisets, L-fuzzy sets etc. Categories and algorithms were taken as (alternative to sets) bases of mathematics. All this was a symptom that there exists a more fundamental structure than set and it may be taken as a basis for mathematics. It appeared that such structure is a named set.

In order to give the exact definition of a named set, let us introduce three collections   Ens, Set, Col . Each of them consists of sets or classes (that are called objects) and their morphisms. Totalities of sets (classes) from   Ens, Set, Col are denoted by   ObEns, ObSet, ObCol,  respectively. Totalities of morphisms (i.e., mappings or binary correspondences between sets or classes) from  Ens, Set, Col  are denoted by   MorEns, MorSet, MorCol, respectively. If   X, Y Î Ob K, then all morphisms from X to Y are denoted by Mor (X,Y) where K is one of the following Ens, Set, Col.

Suppose that the following conditions are valid:

1) ObEns, ObSet Í ObCol;

2) MorEns, MorSet ÍMorCol;

3) the totality   MorCol  is closed in respect to the product of morphisms (it means that if   a,b Î MorCol   and their product ab is defined, then ab Î MorCol).

Let us select some subclass M from the class MorCol. This selection may be to some extent arbitrary that makes possible, using different conditions on M, to define constructions necessary in each concrete case.

Definition 1. A named set (with respect to M ) (or N-set) is a triad  X = ( X, r, I )  where XÎ ObEns,   I Î ObSet,  rÎ Mor (X,I) and  rÎM.

Let X = (X, r, I ) be a named set.

Definition 2. 1) The set  X  is called the support of  X  and is denoted by  S(X);   2) the set  I  is called the name set or the set of names of  X  and is denoted by  N(X);  3) the set   N_f (X) = { a ÎI; (xÎS(X) & (x.a) Îr }  is called the set of nonvoid (factual) names of the named set X;  4) the mapping (correspondence) r is called the naming mapping (correspondence) of  X  and is denoted by  n(X); 5) the element r(x) Î I  (for a single valued mapping r) and the set r(x) = { a ÎI; (x.a)Îr } (for a binary relation or multivalued maping r) is called the complete name of x ÎX  in  X; if r is not single valued, then any element a Î r(x)  is called a partial name of x in  X . Otherwise, it is called simply a name of x.

Many important mathematical notions may be modeled as special cases of named sets as it is demonstrated by the following examples.

1. A multiset is a collection that is like a set but can include identical or indistinguishable elements (Knuth, 1973). For instance, {a,a,b,b,b} is a multiset that contains two elements a and three elements b . Thus a multiset is obtained if we take a named set X and add an axiom demanding that elements from the support S(X) of X are distinguishable if and only if they have different names in X .

2. According to M.Aigner (1979), a multiset on a set S is a function r: S ® N that defines multiplicity of the elements from S (here N = {0,1,2, …, }). Such multisets are the named sets in the case when Ens consists of arbitrary sets and their maps, Set contains the single object N and all binary relations on it, while Col is equal to Ens .

3. In (Hickman, 1980), multisets are defined like in (Aigner, 1979) but instead of N the class Card of all cardinals is taken. So multisets, in this extended sense, are also special cases of named sets.

4. A fuzzy subset A of a universe U is a pair (A, m ) where m is the membership function of A (Zadeh, 1965). If the universe U may be an arbitrary set, then the above definition gives us the general notion of a fuzzy set. Thus fuzzy sets are the named sets when Ens consists of arbitrary sets and their maps while Set contains the single object [0,1] and all relations on it.

5. Taking instead of the interval [0,1] a complete lattice L we receive the notion of L-fuzzy set (Goguen, 1967) or when U = X ´ Y we have L-fuzzy relation (Salii,1965). So, L-fuzzy sets and relations are also special cases of named sets.

Definition 3. A named set X is called: 1) normalized if N_f (X) = N(X); 2) a singlenamed set if N_f (X) consists of a single element; 3) an individually named set if n(X) is a bijection; 4) a one-to-one named set if both sets S(X) and N(X) consist of one element.

Examples of single named sets give us usual sets. Really in the framework of named set theory we can have a new understanding of different trends in the mathematical set theory. The first fact that lies on the surface is that any usual set is a single named set because we cannot speak about, use or construct any set without giving a name to it. This name may be a single sign (M, for example), a logical formula {x Î X; P(x) & (y Î Y ((x,y) Î A ÍX ´ Y)} or an algorithm (some partial recursive function etc.). However, this name always exists. So all elements from the set with this name (M, for example) have the same common name (“an element from the set M”).

Definition 4.   A named set    Y = (Y, b, J)   is called a named subset (a weak named subset) of the named set    X = (X, a, I)   if  Y ÍX,  JÍI and  b = a|_(Y,J)   ( and b Ía Ç (Y ´ J) ). Such relation between named sets is called the inclusion (the weak inclusion) and is denoted by Y Í X   (Y Í_w X).

Definition 5. If   X = ( X, r, I )  and  Y = ( Y, q, J ) are named sets then a morphism from X to Y is a pair   F = (f, g ) where  f: X ® Y, g: I ® J   for which the equality   fq = rg   is valid, i.e., the following diagram is commutative

For morphisms   F = (f,g): X ® Y   and  J = (t,s): Y ®Z, their product (composition) is defined in a natural way as  FJ = (ft, gs): X ® Z.

Theorem 1. If the classes Ens, Set and Col are categories, then the collection NSet of all named sets and their morphisms is also a category (Burgin, 1984).

If we take modern analysis the main structure on which functions are defined and studied are different kinds of manifolds: topological, differentiable, smooth etc. The most general kind is a topological manifold. Each of them is a named set X = (X, r, I) where X is a topological space, I is some n-dimensional vector space R and a is a continious relation that is a local homeomorphism, i.e. for each point x from X there exists an open neighborhood that is homeomorphic to some open subset of R . Conditions that define special cases of topological manifolds (differentiable, analytical etc.) may be also formulated in the language of the named set theory, i.e., as conditions on named sets that are obtained by application to these named sets definite set-theoretical operations.

Scalar, vector and tensor fields on manifolds are also named sets having the form X = (X, n, D) where X is the same as above, D is some set of scalars (real or complex numbers), vectors or tensors and n is a function defined on X.

Categories play an important role in modern mathematics. Any category K consists of two classes  Ob K of objects from K  and  Mor K  of morphisms from  K. For any two objects A and B from  K the set  H(A,B) from the class  Mor K  is singled out. A partial binary composition of morphisms is defined in  MorK. Then to each morphism f from H (A,B) a one-to-one named set ({A}, f, {B}) is corresponded. In such a way, a system T(K) of one-to-one named sets is related to the category K. The system T(K) satisfies three conditions (i) – (iii) (Burgin, 1988). Any system T of one-to-one named sets satisfying conditions (i) – (iii) is called categorical. It is shown that there exists a one-to-one correspondence between abstract categories and categorical systems of named sets. By this correspondence, any functor between abstract categories is mapped into a homomorphism of categorical systems of named sets.

If we take mathematical logic we also see a lot of different named sets. As an example we can take such important construction as model. It should be noted that there exists no generally acceptable and exact definition of a model in mathematical logic. Some authors treat a model simply as a mathematical structure, exactly as a set with a system of relations (may be functions) on it (Malzev, 1970; Shoenfield, 1967). Other authors interpret a model as a pair consisting of a mathematical structure (the same as above) and a partial map of a logical language into this structure (Chang and Keisler, 1973; Mendelson, 1963). It is necessary to note that in the second case we again have some named set. Really, when one speaks about a model of some logical language then he factually bears in mind the named set (L, i, M) of an interpretation of the language L into some mathematical structure M. As mathematical structure M (that is called a model of the language L) the set on which the predicates and functions are defined usually is taken. The map i is built in such a manner that there are correspondences:  (i) between predicate symbols from the alphabet of the language L and predicates having the same number of variables defined on M;  (ii) between functional symbols and functions on M; (iii) between constants and elements of the mathematical structure M .

In a similar way an interpretation of a formal theory (or a deductive calculus) T into a model M is built. The corresponding modeling named set has a form  (T, p, M)  and the truth is such a property that its conservation is demanded from the naming relation p. In other words, if all formulae from T are considered as true, then M is a model of T if and only if the images of these formulae are true in M. It should be noted that when in logic (considered as a model), it is taken a pair consisting of the interpretation map and a mathematical structure, then such model is the part of the modeling named set.

Likewise, any formal calculus  C is also a special kind of named sets. Really, it may be considered as a triad   C = (A, R, T) where  A is the set of axioms,  R are rules of deduction by which from axioms the theorems of the calculus are deduced. These theorems form the set T. The named set  (A, R, T)  is called a named set of calculus rules. The same calculus may be represented by another (deduction) named set  (A, d, T)  where the relation d connects any axiom a from A with such theorems t from T that a is used in a process of deduction of t.

The main progress of contemporary mathematics may be explained as a process of transition from usual (singlenamed) sets to more general cases. It is characteristic for the most mathematical fields. For example, in topology a lot of achievements is connected with the introduction of fibers and their special cases fiber spaces, bundles, smooth fibers etc. But any fiber F is a topological functional named set  F = (E, p, B) ,  i.e., a named set in which the base B and the fiber space E are topological spaces and  p is a continuous projection of  E onto B. It is possible to demonstrate that analogous situations are characteristic for all other branches of mathematics.

Qu’est ce que la Mathesis universalis ?

Ce blog n’a pas pour but les exercices d’érudition, et donc ce ne sera pas non plus le cas de cet article. Il serait pourtant facile, ou plutôt difficile (au sens de “réclamant pas mal de labeur”) de multiplier ici les savants commentaires à propos de cette “chose”, qui apparait au 17 ème siècle chez Descartes (dans les Regulae, ainsi qu’en filigrane dans les trois “songes” de la nuit du 10 au 11 Novembre 1618) , mais qu’il n’ pas inventée (on en trouve trace avant lui, lire par exemple à ce sujet le début du livre “Cet esprit de profondeur” consacré à Wolff par Paccioni aux éditions Vrin, ou bien ce site : http://www.formalontology.it/mathesis-universalis.htm ).

Depuis que je m’intéresse à ce sujet, avec une sorte de fascination pour ce qu’il faut bien appeler la trace, le reliquat, en nos consciences modernes, des projets de Pansophie ou Savoir absolu du Moyen Age latin ou arabe, j’ai multiplié les blogs portant ce nom :

http://mathesis.overblog.com

http://mathesis.blogg.org

http://mathesisuniversalis.blogg.org

http://principiatoposophica.blogg.org

sans parvenir à mieux “cerner” les contours de cette “contrée inconnue”… si bien que dépité par ces échecs retentissants je ne change de thème et ne crée un nouveau blog, de même inspiration philosophique “brunschvicgienne” que les précédents mais avec un nom différent, espérant que l’obsession me serait ôtée :

http://conversionspirituelle.wordpress.com

mais n’est il pas naïf d’espérer sortir d’une impasse existentielle , épistémologique ou “philosophique” en jouant sur les mots ?

Aussi décidé-je de repartir à neuf sur ce blog, nommé derechef “Mathesis universalis”, à partir de ce que j’ai compris, quand même, suite à mes réflexions, c’est à dire pas grand chose !

¨pour tout dire…voici :

il est impossible de “cerner les contours” de la Mathesis universalis, en vue de la définir, de dire “ce que c’est”, parce qu’il s’agit de l’Infini ! ni plus ni moins ! on n’en fait pas le tour , comme parait il certains philosophes affectionnent le “tour du propriétaire dans les jardins du passé”.

Hegel définissait le Temps comme le “Concept existant empiriquement” (“Daseiende Begriff”). D’après Kojève  le “Concept” hégélien est l’unitotalité intégrée des notions précises et non contradictoires formées par l’intellect humain.

Mais je ne me fie ni à Hegel ni à Kojève pour ma recherche, ma vie  philosophique personnelle : c’est Léon Brunschvicg qui m’inspire, et à travers lui Descartes, Spinoza, Malebranche, Kant et Fichte.

Je paraphraserai donc Hegel pour dire une chose entièrement différente mais que je crois vraie, au titre d’une définition-projet dont parlait  Kojève, “définition-projet” que ce blog aura pour travail de “développer entièrement” (ce qui est impossible, s’il s’agit de l’Infini, et il s’agit de l’Infini, sinon “vaine est notre foi”  !!):

“Le Temps est la Mathesis universalis existant empiriquement”

(à suivre)

Philosophie, science, religion

 

 

 

 

 

Ce blog, qui se situe dans la lignée des précédents blogs “Mathesis universalis”, s’inspire principalement de la philosophie de Léon Brunschvicg (1869-1944), et à travers lui des grands philosophes-savants du 17 ème siècle : Descartes, Spinoza, Malebranche, Leibniz.

Il s’agit de rechercher , selon l’acception du Traité de la réforme de l’netendement de Spinoza, le “vrai bien”, c’est à dire selon nous la conversion spirituelle telle que nous la propose la philosophie de Brunschvicg.

Ou, en d’autres termes, l’union à Dieu, procurant une joie continue et souveraine :  non pas au Dieu des religions abrahamiques, mais au Dieu des philosophes et des Savants, Raison universelle des esprits selon Malebranche (nonobstant le fait qu’il était chrétien).

On sait que l’approche de Brunschvicg est modeste, réalisant la véritable “pauvreté en esprit” que proclamait l’Evangile : la philosophie doit suivre le progrès de la science (c’est à dire, selon nous, surtout la physique mathématique), sans la devancer dans une “gnose ésotérique”, qui ne peut être que charlatanerie ou retour aux vieilles superstitions, mais aussi sans prendre (trop) de retard sur elle; or à partir du 18ème siècle, les philosophes ont perdu le contact avec la science car ils ont été incapables de suivre le progrès prodigieux de celle ci… et l’on mesure aujourd’hui les conséquences désastreuses de cet état de fait.