The Notion of Order in Mathematics and Physics. Similarity, Difference and Indistinguishability
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Abstract
The notion of order as a universal and fundamental conceptual category is discussed as being based on sets of similar differences and different similarities. A discussion of relationships between order and disorder is followed by a proposal for a mathematical theory based on non-ordinality which could also have relevance for indistinguishables in physics.
Keywords
Order Similarity Difference Indistinguishability Indistinguishables1 Background
The notion of order as a fundamental concept in physical theories was introduced by a series of pioneering papers published by David Bohm [1, 2, 3, 4, 5]. These ideas and their relation to quantum phenomena were further developed in discussions with Basil Hiley. A comprehensive survey of these ideas were summarised in their classic book “The Undivided Universe” [7]. For the very latest review of some aspects of this work see the review article by Basil Hiley [6]. The wider relevance of these ideas to philosophy and science will be found in David Bohm’s book “Wholeness and the Implicate Order” [8]. It is important to emphasize that these notions were fundamentally seen as dynamic, so ordering would be the general concept while order is a particular static aspect. Implicate order is thus unfolding into an explicate order and the explicate order enfolding into an implicate order. The verbs unfolding and enfolding are stressing the dynamic nature of order and also how these orders are dynamically related in an overall process.
“The basic idea is to introduce a new concept of order, which we call the implicate order or the enfolded order. This is to be contrasted with our current concepts of order which are based on the ideas of Descartes who introduced coordinate systems precisely for the purpose of describing and representing order in physical process. The Cartesian grid (extended to curvilinear coordinates), which describes what is essentially a local order, has been the one constant feature of physics in all the fundamental changes that have happened over the past few centuries. In the quantum domain however this order shows its inadequacy, because physical properties cannot be attributed unambiguously to well-defined structures and processes in space-time while remaining within Hilbert space. Thus, for example, the uncertainty principle implies that it is not in general possible to give a definite space-time order to the motion of a particle in its trajectory.”
D. Bohm took order (and structure) as something more universal and fundamental than most of our basic conceptual categories ([3], p.18). Order is concerned with similar differences, arrangements, organisation and structure. In Bohm’s philosophy all these notions are fundamentally dynamic. The ubiquitous character and significance of order is due to the observation that order is common to all that we conceive and perceive which means relating order both to the order of abstract thought of mind as well as to matter and external reality.
“Our notions of order in physics have generally been tacit rather that explicit and have been manifested in particular forms which have developed gradually over the centuries in a somewhat fortuitous way. These latter have, in turn, come out of intuitive forms and common experience. For example, there is the order of numbers (which is in correspondence to that of points on a line), the order of successive positions in the motions of objects, various kinds of intensive order such as pressure, temperature, colour etc. Then there are more subtle orders such as the order of language, the order of logic, the order in music, the order of sensation and thought etc. Indeed the notion of order as a whole is not only vast, but it is also probably incapable of complete definition, if only because some kind of order is presupposed in everything we do, including, for example, the very act of defining order. How then can we proceed? The suggestion is that we can proceed as in fact has always been done, by beginning with our common intuitive notions and general experience of order and by letting these develop so as to extend into new domains and fields of application.”
As order is seen to be common to all perception and conception it does not seem possible to give it some exhaustive formal or verbal definition. The question is then how can we hope to describe order? The suggestion implied above is that one could start to make explicit what we already know and point to certain essential features of what is mostly tacit knowledge. The situation has some similarities to bare concepts in formal theories such as class, element and morphism in set theory and category theory. Bohm’s suggestion was that the essence of order is based on organisation of similarities and differences. This proposal was based on an analysis of how we construct and form categories both in perception and in abstract thought.
- (i)
A and B are similar relative a context or background,C. Similarity is established when the exchange of A for Bmakes no difference inC which can be written as A:C::B:C.
- (ii)
A and B are similar in relation to a third entityC. This is commonly expressed as A:B::B:C. The introduction of C gives the possibility of creating similarity.
- (iii)
Similarity in a process. Let P denote some process. To be a process something must become different. Consider the process P:A becoming B and B becoming C, where B is different from A and C from B. C could then be similar to A which could represent a process with an aspect of sameness over time.
2 Order and Mathematics
It was Bohm’s contention that also in mathematics order is more fundamental than relationship and class. For instance, when two things are related they are comprehended within some totality of similar or common order, [3], p. 19. Some order is implicit or explicit always in the background which is logically prior to the notion of relationship or context, apparently implying that order is prior to the notion of relationship. Now the question is: How to develop a mathematics based on order (and structure)? This could entail creating a new set of axioms which treat order and structure as basic concepts. This is customary in mathematics exemplified in axiomatic geometry by point and line as bare concepts. Creating such axioms will no doubt involve a considerable amount of development and refinements of ideas and this goes far beyond the beginnings to be discussed in this paper. The main focus here will be on examples of some simple mathematical order, structures in terms of similarity and difference, and finally a first attempt to formalize a basic aspect of difference expressed as indistinguishability.
2.1 Equivalence Classes
- 1.
a∼a
- 2.
a∼b iff b∼a
- 3.
a∼b and b∼c implies a∼c
- (i)
Similarity relative a context, i.e. the equivalence class C. Two objects, a and b, are seen as similar relative C as a and b can be exchanged for each other as representatives of the class C, disregarding all other differences.
- (ii)
The similarity between the objects: The equivalence relation ∼ can be seen as denoting a difference between two objects: a∼b. In this sense all objects in a class have similar or the same difference to each other via the relation ∼. By the reflexion condition (1), we have a∼a, so the difference between a and b is the same as the “difference” between a and a. We could express this as a is to b as a is to a or a:b::a:a. Written in this way it becomes explicit that we “cancel” all differences between the objects in the same class.
Furthermore, the equivalence relation ∼ establishes different equivalence classes. These classes can now in turn be treated as differences. It is now possible to establish similar differences between these classes.
2.2 Sequential or Successive Orders Based on Similarity and Difference
- (i)
A straight line, being of the first class with one independent difference.
- (ii)
A circle, being of the second class with two independent differences.
- (iii)
A spiral, being of the third class with three independent differences.
2.3 An Example from Algebra
Before discussing geometrical curves to illustrate various orders, it might be appropriate to give some simple examples from algebra.
- Ex 1
Logic is clearly an ordered structure. If a structure must have connections between orders, this connection is displayed in a proof in logics where the link or connection is established by rules of necessity i.e. necessity directly connects: It gives necessary connection in necessary relationships.
- Ex 2
Quaternions
Each element i,j and k generates an order of similar differences:
i,i^{2},i^{3},…; j,j^{2},j^{3},…; k,k^{2},k^{3},… where each i,j and k generates an order which is the constitutive difference of respective order. We have the relations: i^{2}=j^{2}=k^{2}=−1;i⋅j=k etc. which shows that i,j and k are connected by the number −1 i.e. two elements of two different orders turn out to be the same and therefore form a link between different orders. This also shows that the parts are connected to each other in an organized system, structure, via some common element etc.
2.4 Orders of Increasing Degree Represented by Geometrical Curves
2.4.1 Orders of First Degree: A Straight Line
2.4.2 Orders of the First Degree, Second Class: A Circle
We still have a:b::b:c etc. Note that there is only one similarity or ratio, which could be denoted S^{1}=(a,b,c,…)≡(D_{1},D_{2},…) and each difference D_{k} consists of two constitutive differences (d_{1(k)},d_{2(k)}) where d_{1(k)} represents difference of position and d_{2(k)} represents difference of direction.
2.4.3 Orders of the First Degree, Third Class: A Spiral
Note that we still have only one similarity as the lengths of the segments are progressively decreasing. However the similarity (or ratio) between successive steps remains invariant. Independence in this context only means that one difference cannot be derived from the other.
2.4.4 Orders of the Second Degree, First Class
“Thus far we have considered various kinds of similarities in the differences, to obtain curves of the first, second third classes, etc. However, in each curve, similarity (or ratio) between successive steps remains invariant Now we can call attention to curves in which this similarity is different as we go along the curve. In this way, we are led to consider not only similar differences but also different similarities of the differences.”
We can now consider the difference of successive similarities (S^{1},S^{2},S^{3},…) as a second degree of difference. From this, we can develop a second degree of similarity in theses differences. S^{1}:S^{2}::S^{2}:S^{3}”, [8], p. 116. This is the beginning of generating a hierarchy of similarities and differences, i.e relating lower levels of order to describe order of orders.
By considering geometrical curves involving differences and similarities of a third, fourth, and so on degree, Bohm introduces the concept of what he calls orders of higher degree.
D. Bohm’s own view of the relationship between order and mathematics was essentially that the current aspects of order in mathematics were insufficient. So for instance, the order relation < tacitly presupposes that each element is considered as completely constituted, i.e. the order of elements is external to the elements themselves, referring to how the elements are different from each other and then being related. This is thus contrasted by the above discussion where the order of differences of the chords, etc actually was seen as constituting the curve, not only describing it. The more general challenge as Bohm saw it was ([3], pp. 23–24):
The hierarchical potentialities of order.
Not assuming that a mathematical structure is composed of separate elements where orders and relationships are external to what these “elements” are.
Explicitly differentiate between constitutive and descriptive order/difference. To be able to exhibit how these orders are related in a vast set of cross-references of one aspect of structure to another.
3 Order–Disorder
The notion of absolute disorder is a non-category. Some kind of order is common to everything we can perceive and conceive. There is also a general problem in defining a fundamental concept (like disorder) only by negation. The reason is that the core meaning of a definition is to keep something within certain limits. This is not equivalent to a definition by negation as this means going outside these limits. It is not a problem when the definition is made in a limited context. But universal and basic concepts are not in a limited field and a negation will be without limits: saying disorder is everything which order (which notion itself cannot be exhaustively defined) is not which makes it unclear what is being referred to, [4], p. 308. Furthermore, to define absolute lawlessness and featurelessness or disorder positively as well as negatively we will inevitable refer to some kind or order, rule, law, feature etc. which leads to the absurd concept of law of lawlessness, feature of featurelessness etc. which makes it contradictory. Instead we can talk about relative disorder, relative non-regularity etc. We are free to talk about order of an indefinitely high degree, or of infinite degree. Instead of equating randomness with absolute disorder, we can approach this concept differently by referring to orders of infinite degree. Disorder in the “absolute sense” can be exchanged with an order of infinite degree, from which we in principle could abstract any suborder or particular order, [11], pp. 137–141.
3.1 Related Orders: Causality–Determinism
When two orders are related and when given one, we can more or less determine the other. This has been a necessary condition for prediction and causality in science. Prediction is based on the relation between the order of physical events and the order of time. Long before modern science, astronomers were able to relate the positions of planets, moon etc with the clock-time order and from this the time order make predictions of future positions. Determinism and causality are based on the fact that two orders determine each other. In physics determinism as predictability is based on a close relationship between the sequential order of time and an order of physical events. Seen in the general context of order, predictability is then a particular relationship between the order of time and events, where a few steps in time determine the whole order of events. The concept of implicate order as proposed by Bohm can be regarded as such an example where a basic order of physical processes is not related to time-order but to an order of transformations of an algebra, denoted as an order of enfoldment.
3.2 Unrelated Order: Relative Disorder and Randomness
A classical example of a random series based on physical events is the flipping of coins. The relevant two orders are: The succession in time of throws and the outcomes. These two orders are almost independent which means that given the time-order, one cannot determine the throws and given the throws one cannot determine the time-order. This is what is called a random sequence. This does not mean complete disorder relative time order because the outcomes are not irregular in respect to initial conditions, of position and velocity of the coin. Even if given a series of random events with no known law like dependence on anything else, two such series would be distinguished from one another. In this elementary sense they have some order. The point is that total lack of order has no real meaning.
3.2.1 Unrelated Orders
In statistical physics, thermodynamics, the movements of the individual particles depend on a very large number of other particles. There is almost a complete unrelatedness between the individual particles, the micro order. This independence is a necessary condition for the application of probability theory which in turn gives rise to a statistical relationship, or a macro order which is expressed by laws like pV=nRT which represents a large scale order relating pressure, volume and temperature of a gas. This order is of a much lower degree than the microscopic order of movements of independent molecules and almost insensitive to the order of the microscopic level and where the order of time has disappeared completely. This illustrates that in the limit of large numbers, random orders can approximate regular and simple orders of low degree. In the given example the gas law does not even make sense at the microscopic level. Another example related to physics is Brownian motion. This case illustrates that the independence of the micro order and the macro order is only relative. A small particle is exposed to random collisions. The statistical analysis (random walk) gives that the distance from some original position is proportional to the time passed times the square root of time: \(\mathit{constant}\cdot\sqrt{t}\). This means that we have not lost a relation to time-order completely and we can use this statistical relation for predictions. Therefore this example displays a situation somewhere in between complete loss of a relationship between a microscopic order, appearing in the dynamics of the particles, and macroscopic order.
3.2.2 Disorder and Randomness; Context Dependence
Relative disorder is not only to be found in physical events. As a simple illustration, take the square root of 2. Express \(\sqrt{2}\) as a string of symbols, representing the digits in some base. If now part of this string is presented to someone who does not know its origin, the person would find it fulfilling all criteria of randomness and “lack of order”, and as a consequence could in no way predict the next digit from the previous ones. In the context, when the appropriate algorithm is given, it is perfectly determined. So we have clearly a context dependence of randomness.
An analogous situation is the generation of random numbers by a computer. In a context that does not include the computer and the algorithm, the number series produced will clearly exhibit randomness. Again in a context including the program the numbers are generated by an algorithm of a quite low degree of order. The two notions order and disorder have been seen to be not mutually exclusive but rather complementary. Science is in fact exploring new orders between chaos and simple kinds of order as is seen in turbulence, chaos theory, fractals etc. Prigogine has demonstrated that regular orders can emerge in certain chaotic systems. This means that these systems must move through a whole spectrum of orders between chaos and regular orders of quite low degree. Furthermore, Mandelbrot has shown the existence of set with fractal dimensions, between spatial dimensions. Such sets can be of almost infinite complexity of almost infinite degree of orders as exemplified by Julia sets. This situation is also reflected in modern chaos theory. Incorporating order as a fundamental category in our thinking could give new perspectives to perceive and conceive new laws and regularities in Nature.
3.3 Indistinguishables: Different but not Distinct
We have now been discussing mainly sequential orders. What could be said about non-sequential orders? The most basic sequential order is based on differences which can be ordered or indexed, i.e. having ordinality in a mathematical or set theoretic sense. This way of thinking leads rather directly to a notion of non-ordinality or indistinguishability.
The motivation behind studying indistinguishables is here three-fold. As mentioned earlier D. Bohm suggested that order is basically described as a set of similar differences and different similarities. Difference is seen as more basic than similarity. The reason is that similarity presupposes difference which makes difference logically prior to similarity. In fact, similarity is a consequence of disregarding difference. In such a context difference becomes fundamental. It is therefore natural to ask if there are different kinds of basic differences, i.e. is there really only one difference, usually expressed as in a≠b? It is conceivable that two different objects comprise of two aspects of difference: one collective and one individual. The collective aspect refers to some collective totality, whereby different objects are different because they are differently contributing to the whole, or collective. One could say that each object is defined collectively by being different from all others in the shared context or collective. The individual difference then concerns a direct relation between two individuals. This difference is always used when some object is named, labelled, indexed to identify each object uniquely. A “collective” difference is then reflecting that objects are different in the sense that they, by the very being part of some whole or collection, are differently contributing to this whole. If they were not, they would not be different at all. If cardinality represents the collective aspect of difference, ordinality would represent the individual. It is hard to see any reason why these two aspects necessarily should be identical. This motivates a proposal of two basic kinds of differences where non-ordinality will imply indistinguishability. Another reason for a discussion of indistinguishables is that there are very few systematic attempts to deal with id:s (here after I use the shorthand id for indistinguishable). Indirectly, any limitation in the scope of the subtlety of mathematics could entail limitations in aspects of our understanding of reality. Thirdly an obvious example of collective difference but not of individual difference comes from physics. In quantum theory we have the notion of Bose-Einstein statistics describing bosons, which are treated as indistinguishables. Moreover Fermi-Dirac statistics deals with indistinguishable fermions. Even in “classical” physics the notion of id turns up in discussions of Gibbs paradox in the context of statistical thermodynamics. There is no basic mathematical treatment available in these situations but rather general rules of thumb.
- (i)
The introduction of a third pairity relation between objects (to differ from multiset theory), denoted as α⊻β.
- (ii)
Relaxing ordinality, necessarily implies that such objects cannot be individually labelled, named or indexed.
- (iii)
Taking the “collective” aspect of difference, as fundamental. This means that the objects are differently contributing to the cardinality of a class, but not to the ordinality. Differently put: Each object contributes to the whole by contributing (differently) to the cardinality of some class where the class represents a “whole”.
The original source of inspiration was Parker-Rhodes work on id:s in [10]. Many ideas and notions in this discussion derive from this work. However, the path taken here is not faithful to that theory being triparitous at a semantic level, which builds on a non-transparent notion of information.
[Some notations: In α⊻β, α and β are called id:s. A class containing id:s is called a Sort denoted as S. To avoid confusion with Set-theory “β in S” means β belongs to S.]
4 A Novel Parity Relation ⊻, Representing a New Difference Between Objects Which Lack Ordinality
4.1 Preliminaries
Parity relations
Definite parity relation | Negations |
---|---|
a=b, equal | a≠b, a and b are different |
a⊻b, a and b are id:s | a | b, a and b are distinguishable |
a⊥b, a and b are distinct | a † b, a and b are indistinct |
All relations are symmetrical. It should be emphasized that while being triparitous a theory of id:s is fundamentally different from a many-value logic on one hand and from a theory of multisets (allowing for a number of identical elements in a set) on the other. This will become more clear below. Each negation of =, ⊻ and ⊥ is defined as the disjunction of the other two. This is important to stress as, for instance a † b could mean that neither =, nor ⊻ can be excluded.
Below follows an outline of a theory of a (quasi) class of indistinguishables (id:s). It should be pointed out that a formal Sort-theory is still under construction and the axioms below can be subjected to modifications at a later stage. These (quasi) classes are called Sorts and have members and parts that might be id:s. Parenthesis are used to denote contexts, such as \([\mathcal{A}]\) which can be read as “in the context of \(\mathcal{A}\)”. A “collection” is a specific context, written as (A) and provides to some extent an analogy to {M} in set theory. A collection, (X), has the main function of “enclosing” which is inside, entailing a separation from what is outside, thus creating a collective context or a whole. A part of a Sort is called subsort or member.
Definition 4.1
(Sort)
- (i)
A Sort is a (quasi) class of objects which might contain id:s.
- (ii)
A Sort is called pure iff a ⊥ b does not hold for any two objects in the Sort. Otherwise it is named mixed.
Axiom 4.1
(Existence of Sorts)
∃x, ∃yand ∃S (Sort) such thatxinS&yinS, wherex⊻y.
Axiom 4.2
(Axiom of cardinality)
Every Sort is associated with a unique cardinal number, which is identical to a natural number.
Axiom 4.3
(Pairity relations of Sorts)
- 1.
Two Sorts, or their parts or members, must have one of the possible six pairity relations.
- 2.
SandTtwo Sorts: if there is no objectαinSandβinTsuch thatα⊥βand ifSandThave equal cardinality, eitherS=TorS⊻T.
Comment: If for some α in S and some β in Tα⊥β, this will imply that S and T can be indexed by α and β respectively and can thus be ordered, hence S⊥T.
Axiom 4.4
(Axiom of extensionality)
(∀z;z in S⇔z in T)⇔S=T. Sis equal toTiff any item inTis an item inSand vice versa.
Axiom 4.5
(Collection Axiom)
- (i)
χ≠(χ), except when cardχ=1 or cardχ=0.
- (ii)
cardχ= card(χ), unlessχitself a collection.
- (iii)
((χ))=(((χ))).
- (iv)
GivenxinA, and someyin (A) it is generally undecidable whetherx=yorx⊻y, without further information.
Axiom 4.6
(Identity Axiom)
χχ=χ (i) andχ=ψiff cardχψ=1 (ii).
Definition 4.2
(Empty Sort)
An empty Sort ∅, for any x ⇒x not in ∅.
Definition 4.3
(Singleton)
A singleton is a Sort, s, for which there is only one object in s: x in s and y in s⇒x=y.
Axiom 4.7
(Existence of singletons)
- (i)
cards= card(s)=1.
- (ii)
s=(s).
Comment: The second criteria makes Sort-theory very different from set theory where members and sets are clearly separated, M≠{M}.
Axiom 4.8
(Existence of Empty Sort)
There exists an empty Sort denoted ∅ according to the Definition 4.2 such that card∅=0 and it holds that: ∅=( ) and (∅)=( ).
Comment: The uniqueness of ∅ follows from Axiom 4.4 (axiom of extensionality) by assuming ∅_{1} and ∅_{2} and observing ∅_{1}≠∅_{2} cannot be falsified.
Postulate 4.1
An element or set in the category of Set can only have the parity relation ⊥ (distinction) with an object in a pure Sort.
Postulate 4.2
GivenaRbandbRcwhereRis one of the parity relationsR∈{=,⊻,†}. It can never be the case that any pair ofa,b,cis distinct (such asa⊥c).
Postulate 4.3
GivenaRbandb⊥cwhereR∈{=,⊻,†} implies distinction betweenaandc, i.e. a⊥c.
Postulate 4.4
(Invariance to permutation of id:s)
In any relation or expression where two id:s are permuted or interchanged, provided that there is no ambiguity that the same token, sayx, oryrefer to the same object, for every occurrence of the same, the relation or the expression remains invariant.
Comment: “Postulate” reflects a more temporary status than “Axiom”.
4.1.1 Id:s Form Equivalence Classes of Sorts
From the transitivity implied by Postulate 4.2 it is seen that the pairity relation a†b is an equivalence relation. It holds that: a†a (as † is the disjunction of ⊻ and =) and also: a†b and b†c⇒a†c as a⊥c is impossible due to Postulate 4.2. The term “class” is not used in a set-theoretic way. Moreover a in E_{1} and b in E_{2} where E_{1} and E_{2} are two different sort-classes of id:s. The only possible relation is a⊥b as otherwise a and b would be related by = or ⊻, i.e. they would be equivalent.
Definition 4.4
The equivalence sort-classes of id:s define pure Sorts.
4.2 Introduction of “Ordered” Pairs; Non-Ordinality
The basic ordering of two objects is commonly defined as an ordered pair P(a,b)=(a,(a,b)). The parenthesis ( ) denote a collection of items as defined above i.e. a Sort seen as a collective whole. To be consistent with a notion of non-ordinality as formulated in the axiom above, the natural choice is P(a,b)=P(b,a).
Axiom 4.9
(Non-Ordinality)
P(a,b)=(a,(a,b))=(b,(a,b))=P(b,a).
4.2.1 Fundamental Ambiguity
Consider P(a,b)=P(b,a) written as (a,(a,b))=(b,(a,b)). If the first objects a and b in respective pairs would be well defined it would imply a=b. To avoid this an alternative is to say that the first object is not well defined. The proposal here is that the first element only represents one of the two elements in (a,b), and does not refer to a particular one. To be consistent with this approach it is further assumed that in expressions like ((a,b),(a,b,c)) we can only say that the first “collection” of objects appears in the second (a,b,c) and cannot make a difference between this expression and the expressions ((a,c),(a,b,c)) or ((b,c),(a,b,c)). To deal with this kind of ambiguity a special quantifier \(\hat{U}\) is called for: \(\hat {U} \ a \ b \ c\) , meaning one of a,b,c but no one in particular. This will be discussed separately below.
4.2.2 Context Dependence
The ambiguity caused by the non-ordinality axiom P(a,b)=P(b,a) (Axiom 4.9) means that in general one cannot demonstrate that two identical symbols refer to the same object like the two symbols a in P(a,b)=(a,(a,b)) appear in different contexts. When the contexts are different there is no guarantee that we can unambiguously refer to the same item or object. To have a consistent theory, this is then generalized: Given some collection (α,β,γ) of id:s. If now one or more are duplicated or “taken out” of the collection these symbols cannot unambiguously refer to the same objects as being in the original collection. To bring this out more explicitly: Consider the “collection” C of unordered pairs: C=((x,z),(y,w)). Now permute x↔y giving C′=((y,z),(x,w)) which by the Postulate 4.4 leaves C unchanged that is: C=C′. If we now try to identify say x and z in both C and C′ i.e. set x=z in both C and C′. We would get ((x,x),(y,w)) for C, and as (x,x)=(x), according to the identity axiom (Axiom 4.6), we have the equality: ((x),(y,w))=((x,y),(x,w)) (for C′), which obviously is false. The mistake is to identify two elements x,z being in different contexts or collections and having different pairity relations in C: x⊻z while in C′: x†z. This is because x and z are in different contexts in C′, i.e. the contexts of (x,w) and (y,z) respectively. Without further information it could be that x=z and hence we can only say x†z. This kind of context dependence calls for particular rules when we can relate two id:s.
4.3 Existential and Universal Quantification
4.3.1 Existential Quantification
In a standard approach existential quantification of a predicate is generally expressed as ∃x; Qx is true, where x belongs to some class C and asserts that there are some members in C which have the property Q. It is assumed that it is possible to select the appropriate items by some (unspecified) procedure. However, in the case of Sorts, the nature of id:s does not allow for selecting a particular item as this would imply some labelling, naming or indexing. Even to say that some particular item in a Sort has the property Q is to assert something that cannot be proven by any algorithmic approach. By testing one by one we cannot identify if one item has been tested or not. To assert that some member of a Sort has the property Q is something that cannot be disproven. If we test some x in a Sort to see if Qx is true and find that it is not, we have to go on testing every member in S. However, we cannot label which ones have been tested so we could end up in indefinitely examining the same individual. There is however one exception. In a definition one could demand that something having a property Q shall exist. Such a statement is not open to test. However, it may turn out in some other context that the definiend is proven not to exist if due care is not taken. Generally this is avoided by explicitly only taking the cardinality of the individuals having property Q. More precisely: as any Sort has a unique cardinal number one should be able to assert that in a given Sort a certain subsort with definite cardinality has the property Qx, as the subsort is only specified up to cardinality.
4.3.2 Universal Quantification
The situation is different regarding universal quantification i.e. using all, every or any. As no particular selection is involved, universal quantification does not create problems. We can write this as: ∀x; x in C⇒Px=( ) where C some (Sort) class. No testing procedure is required in universal quantification: we can arbitrarily pick any one.
4.3.3 Conditional Quantification
4.3.4 Negation of Universal and Conditional Quantification
The negation of a universal quantification: ¬(∀x;x in C⇒Px=( )) “destroys” the universal quantifier ∀ as it means that there are somex in C for which Px is false, or (not P) is true, turning the statement into an existential quantification. This does not mean that we cannot negate P, but we have to use some conditional quantification ∀x;Qx=( )⇒(not P)=(/), which says that P does not hold for anyx such that Qx is true. This discussion is now expressed in the following axioms.
Axiom 4.10
(Forbidden existential quantification)
Unconditional existential quantification is forbidden, except when demanded so by (reasonable) definitions. Reasonable should be read as non-inconsistent or non-incoherent.
Moving on to the negation of conditional quantification to consider ¬(∀x; Qx⇒Px). This means that there exists some x for which Qx is true, but Px is false i.e. (∃x; Px=(/) and Qx=( )). This is existential quantification and forbidden. This does not mean that “not P” (¬P) is meaningless. We can say that ¬P does not hold for anyx such that Qx is true. As “any x” is universal quantification it is allowed for and we write ∀x; Qx=( )⇒¬Px=(/).
Axiom 4.11
Universal quantification is allowed for.
Axiom 4.12
Conditional existential quantification is generally allowed for: Pholds for anyxsuch thatQxis true.
Axiom 4.13
(Existential quantification involving cardinality)
∃xinS; card(xinSforwhichPxistrue)=n, n∈Nandn ≤ card S.
This defines a subsorthofS: \(\forall x \ \mathrm{in}\ h \Rightarrow P \ x \mathrm{\ and\ card}(h)=n\).
5 Subsorts, Unions and Other Things
A subsort of a Sort a part of S which is defined as usual:
Definition 5.1
(Subsort)
h⊆S⇔∀x(x in h⇒x in S) and cardh≤ cardS.
Definition 5.2
(Proper subsort)
\(\exists x \ \mathrm{in}\ S\mbox{ and $x$ not in }h\)and cardh< cardS. (Due to the existential quantifier ∃ we demand the existence by definition.)
- 1.
Writing x in h, y not in h says that only one of two elements x,y,((x,y) in S), one belongs to h, the other does not. It is not meaningful to assign a certain property to x in particular (in this case “x in h”) as this would entail a “labelling” of x and y which is forbidden. Consequently x in h,y not in h cannot meaningfully be distinguished from y in h,x not in h.
- 2.
cardh_{1}≠ cardh_{2}⇒h_{1}⊥h_{2}; (see Axiom 4.3); The reason is that h_{1} and h_{2} can be assigned two different unique cardinal numbers.
- 3.
cardh_{1}= cardh_{2} gives without further information the pairity relation h_{1}†h_{2} as h_{1} and h_{2} have equal cardinality. If they are known not to be identical we will get h_{1}⊻h_{2}.
- 4.
Given a Sort S, cardS=N each n<N will entail an equivalence (Sort) class of subsorts each with cardinalityn. It is seen from this that we get N distinct classes of subsorts of S, one for each cardinality.
As cardinality is fundamental we state:
Theorem 5.1
Given a subsorthof a (pure) SortS, h⊆Sit holds that: cardh= cardS⇔h=S.
Proof
(⇒): From Axiom 4.3 cardS= cardh⇒h⊻S or h=S. If h⊻S we can permute h and S in h⊆S i.e. S⊆h and according to Axiom 4.4 it follows h=S. The implication in the other direction is immediate. □
5.1 Relative Complement
Given h, a subsort in S we need to construct the complement of h in S. We do this by the following
Definition 5.3
(Relative Complement in S)
- (i)
cardh+ cardCh= cardS.
- (ii)
∀x in S it holds that x in h or x in Ch.
- (iii)
From (ii) follows immediately that C[Ch]=h. Consider Ch and C[Ch] where every x either in Ch or C[Ch], so if x not in Ch⇒x in C[Ch]. While (ii) gives also x in h. If x in h⇒x not in Ch and thus x in C[Ch] which gives the equality (Axiom of extensionality).
5.2 Intersection of Sorts
Definition 5.4
(Intersection of Sorts)
The intersection of S_{1} and S_{2}: S_{1}∩S_{2}⇔x in S_{1} and x in S_{2}.
5.3 The Union of Sorts
Considering the usual definition:
Postulate 5.1
- (i)
xinS_{1}∪S_{2}⇔x in S_{1} or x in S_{2}. This reveals a difficulty: If we want to identify the members inS_{1}∪S_{2}there is no clear way to associate an elementx uniquely withS_{1}and not withS_{2}. This would imply a labelling of the elements: sayxinS_{1}, xnot inS_{2}whileynot inS_{1}, yinS_{2}. This is indistinguishable fromxinS_{2}, xnot inS_{1}andyinS_{1}, ynot inS_{2}. Therefore (i) calls for a modification: It is clear thatS_{1}∩S_{2}and cardS_{1}∩S_{2}does not create the same problem because every elementxinS_{1}∩S_{2}has the same “label” namely “xinS_{1} and xinS_{2}” or is indexed by cardS_{1}∩S_{2}. Using intersection and relative complement, union can be expressed and defined by: S_{1}∪S_{2}=C[CS_{1}∩CS_{2}] whereS_{1}andS_{2}belong to the same equivalence class (Sort), andCS_{i}denotes the complement ofS_{i}inS.
- (ii)
S_{1}∪S_{2}=C[CS_{1}∩CS_{2}].
- (iii)
cardS_{1}∪S_{2}= cardS_{1}+ cardS_{2}− cardS_{1}∩S_{2}.
Now, take the complement of each side of (ii): CS_{1}∪S_{2}=CS_{1}∩CS_{2}, as CCS=S. For any x in S_{1}⇒x not in CS_{1}⇒x not in CS_{1}∩CS_{2}⇒x not in CS_{1}∪S_{2}⇒x in S_{1}∪S_{2} and analogously for x in S_{2}. This demonstrates that we can work comfortably with the usual definitions of union under the restrictions given. This also gives S_{i}⊆S_{1}∪S_{2}, (i=1,2).
Theorem 5.2
cardS_{1}∩S_{2}= cardS_{1}if and only ifS_{1}⊆S_{2}
(⇒): cardS_{1}∪S_{2}= cardS_{1}+ cardS_{2}− cardS_{1}∩S_{2}= cardS_{2}. Assume the contrary: ∃x;x in S_{1} and x not in S_{2} implying cardx∩S_{2}=0. x in S_{1}⇒x in S_{1}∪S_{2} as well as x∪S_{2}⊆S_{1}∪S_{2} (by definition of union). Hence, cardx∪S_{2}≤ cardS_{1}∪S_{2}, as x∪S_{2} is a subsort of S_{1}∪S_{2}. But then cardx+ cardS_{2}− cardx∩S_{2}≤ cardS_{1}∪S_{2} i.e. 1+ cardS_{2}≤ cardS_{2} which is a contradiction ⇒ for all x in S_{1} it holds that x is in S_{2}.
(⇐): x in S_{1}⇒x in S_{2}⇒x in S_{1}∩S_{2}⇒S_{1}⊆S_{1}∩S_{2}⇒ cardS_{1}≤ cardS_{1}∩S_{2} and obviously: cardS_{1}∩S_{2}≤ cardS_{1} which implies cardS_{1}= cardS_{1}∩S_{2}. The theorem gives immediately that cardS_{1}∩S_{2}= cardS_{1}= cardS_{2}⇔S_{1}=S_{2}.
5.4 Partitioning of a Sort
Definition 5.5
- (i)
x in h, h subsort of H⇒x in S.
- (ii)
Any two subsorts h′,h″ of H⇒h′∩h″=∅. (∅ is here the empty sort.)
- (iii)
cardh_{(1)}+ cardh_{(2)}+⋯+ cardh_{(k)}= cardS.
- 1.
h′, h″ subsorts of H: cardh′∩h″= card∅=0.
- 2.
S=⋃h_{(k)}.
From Axiom of cardinal extensionality it follows that ⋃h_{(i)}=S. Note that the partition defines a multiset { cardh_{(1)},…, cardh_{(k)}}. We need further to postulate the existence of partitioning.
Postulate 5.2
(Partition postulate)
Given any Sort, Swith cardS=N. For every integer partitioning ofN={n_{i}}, there is a corresponding partitioning ofSinto subsortsh_{(i)}, where cardh_{(i)}=n_{i}such thatS=⋃h_{(i)}andh_{(i)}∩h_{(j)}=∅ forh_{(i)}≠h_{(j)}.
As has been observed before, such a partitioning is only unique up to cardinality, any permutation of id:s between the subsorts leaves the partitioning invariant.
6 Blur: A Novel Ambiguity Quantifier
No property of id:s is more basic than being subject to confusion. As was seen from discussing “ordered pairs” one would need some consistent notation to deal with situations like P(a,b)=(a,(a,b))=P(b,a)=(b,(a,b)), where the first element does not refer to a particular element in (a,b) but to one of the two elements. Given (a,(a,b)) we want to express something that represents one of (a,b), but indecisive which, and consequently not the two taken together as a collective. This is written as \(P(a,b)=(\hat{U} \ a \ b,(a,b))\) where \(\hat{U} \ a \ b\) denotes and represents one of (a,b) but not one particular. If we exclude distinction, a⊥b, the only possible pairity relation between \(\hat{U} \ a \ b\) and the elements a,b could be: \(a \dag\hat {U} \ a \ b\) and \(b \dag\hat{U} \ a \ b\). To restrict \(\hat{U} \ a \ b\) to the collection (a,b) we would also need to say that \(\mathrm{card}(a,b,\hat{U} \ a \ b)=2\). We will now formulate a definition of \(\hat{U}\).
Definition 6.1
(Blur \(\hat{U}\))
- 1.
\(P\ \hat{U}xy\Rightarrow Px\) and Py.
- 2.
Px and \(Py\Rightarrow P\ \hat{U}xy\).
6.1 Some Relations Involving Blur
Definition 6.2
\(\hat{U} \hat{U} = \hat{U}\).
For a union \(z\ \mathrm{in}\ \hat{U}h \cup\hat{U}g\) could be written ¬(z not in h∨z not in g)=¬(z not in h∪g).
Definition 6.3
\(\hat{U}h \cup\hat{U}g = \hat{U} h\cup g;\hat{U}h \cap\hat {U}g=\hat{U}h\cap g\).
As an illustration we evaluate \(\hat{U} | \hat{U}h \cup\hat{U}g |\): from the Definition 6.3 \(\hat{U}|\hat{U}h \cup\hat{U} g|=\hat {U}|\hat{U} h\cup g|=\hat{U}\hat{U}h\cup g=(\mbox{by Definition~\protect{6.2}})=\hat{U}h\cup g\). Applying associativity gives \(\hat{U}a \hat {U}ab = \hat{U} \hat{U}aab = \hat{U}ab\) which also is easily generalized.
6.2 Distributivity
Distributivity
\(f[\hat{U}xy]= \hat{U} fx fy\).
6.3 Complement of Blur
- 1.
\(C \hat{U}xy=\hat{U}CxCy\), given that cardx= cardy=1.
- 2.
\(\mathop{\mathrm{card}}\hat{U}Cxy=\mathop{\mathrm{card}}S-1\).
7 The Collection Operator
A collection is a class/subtotality, C, which defines a context of its own. This concept is illustrated by comparing a subsort h≡S (S is a pure sort for the sake of argument), and the collection of h, written as (h). Taking any object in h, say α and any object β in (h) we cannot without further information decide whether α≠β or α=β. The collection, ( ), separates C from whatever is “outside” in this respect. To copy or duplicate some subsort E by selecting objects (id:s) is not possible due to the problem with existential quantification. As id:s can not be labelled or indexed in order to know which of the objects have been copied and which have not. This calls for some procedure corresponding to “sorting out” or “collecting” id:s. One way to deal with this is to define an operator which acts as copying and “collecting together”. By letting such an operator Ω act on some sort S, we distinguish some collection from the sort. If h is a subsort of S, we write Ω_{h}S≡(h). This means that Ω_{h}S=(h) defines a duplication of the objects in the subsort h in S. One could say that Ω_{h} is copying and “lifting out” part of S to form a Sort denoted by (h). By the operation Ω or ( ), the items “lose contact” with the original Sort. A consequence of this is that in aΩabc⋯ or a,(a,b,c,…) the two symbols a do not necessarily refer to the same item. This is axiomatised as
Axiom 7.1
(Axiom of Collection Operator)
A collectionΩ_{h}Salso denoted (h) is an operation on the Sort which duplicates and separates a subsorthfromSin the sense that anyainSandbin (h) will generally have the pairitya † b.
- (i)
Ω_{h}S=(h)
- (ii)
Ω_{k}(Ω_{h}S)=Ω_{k}(h)=((k))
- (iii)
cardx=1⇒Ωx=x=(x).
This implies directly that Ω_{k}Ω_{h} and \(\varOmega_{h}\varOmega _{h}=\varOmega_{h}^{2}\) in particular produces a singleton ((k)) or ((h)) as these collections have only one member, (k) and (h) respectively. We can now operate once more Ω((h)) which can only yield the collection of (h) i.e. ((h)) in accordance with (iii). We can therefore unambiguously write \(\varOmega_{h}^{3}S=\varOmega_{h}^{2}S\) or simplified as \(\varOmega_{h}^{3}=\varOmega_{h}^{2}\). These axioms imply: that we do not necessarily get something different when writing \(\mathcal{X} \rightarrow(\mathcal{X})\), i.e. allowing for \((\mathcal{X})=\mathcal {X}\) while demanding that \((((\mathcal{X})))=((\mathcal{X}))\).
8 Functions to and from a Sort
We will consider two kinds of functions: collective functions (to be defined) and functions as operators.
8.1 I Functions as Collection of Pairs: “Business Not as Usual”
A function can generally be defined as a new class of ordered pairs from a direct product, D×R, where D is the domain and R the codomain, with the first component in the pair only occurring once in this class. When sorts are considered special problems arise for two reasons: firstly the id:s cannot be indexed in any unique way and secondly their context dependence described above.
8.2 II Collective Functions
We need some approach where the collective nature of the mapping is basic. A function here called collective function from a Sort S, does not act separately on a particular argument but “simultaneously” on every id and subsort of S. We therefore postulate that the function F, representing a general collective mapping, i.e. from Sort to Sort, from Sort to Set or from Sort to mixed Sort defines a context from the whole domain. The arguments in F are directly related in an image of parts/subsorts of the domain: Given Fα and Fβ, αandβcan be regarded to be in the same context which means being in the same context of the collective mappingF. The underlying idea is to define functions from and to a Sort as a collective operation acting on the whole domain “simultaneously” in contrast to sequentially on the arguments. This approach requires some further definitions and postulates.
Postulate 8.1
(Collective functions)
The arguments in a collective functionFcan be given a definite pairity relation without ambiguity i.e. αandβinFα,Fβcan consistently be seen as different i.e. α⊻β.
Note that this is in contrast to a collection of pairs like [(α,x_{1}),(β,x_{2})] where we could only say α†β where α and β are “copied” from some Sort or collection into the context of the collection of pairs. The Postulate 8.1 states that a collective function defines a common context for objects in the domain to yield definite pairity relations between them.
Definition 8.1
(Definition of Collective Functions and Existence Postulate)
- (i)
F[h_{(1)}∪h_{(2)}]=Fh_{(1)}∪Fh_{(2)}.
- (ii)
F[h_{(1)}∩h_{(2)}]=Fh_{(1)}∩Fh_{(2)}.
- (iii)
cardFh≤ cardh, where h, h_{(1)} and h_{(2)} are arbitrary id:s or subsorts (or elements or subsets if D is a set) in the domain, D, of F.
- (iv)
cardx=1, where x is an element in D.
- (v)
cardFh=0⇔h=∅ for which F∅=∅.
Definition 8.2
(Surjective functions)
F is surjective (onto D) if: cardFD≥ cardR, where D is the domain and R is the codomain.
Definition 8.3
(One-to-one functions)
F is one-to-one if: cardFD= cardD= cardR, where D is the domain and R is the codomain.
From the definitions a number of direct consequences follows:
Theorem 8.1
A partitioning of the domainD=Uh_{(i)}induce a partitioning of the codomain, R, by the image ofFofR: FR, thatFis onto (surjective). This induced partition isUFh_{(i)}=R.
Proof
Given h_{(j)} and h_{(k)} in some partition D=Uh_{(i)} gives Fh_{(j)}∩Fh_{(k)}=F[h_{(j)}∩h_{(k)}]⇒ cardFh_{(j)}∩Fh_{(k)}= cardF[h_{(j)}∩h_{(k)}]≤ cardh_{(j)}∩h_{(k)}=0 as h_{(j)}∩h_{(k)} is empty by definition ⇒F[h_{(j)}∩h_{(k)}] is empty. Moreover, FD⊆R and FD= cardR by definition of an onto-function which gives that FD=R by the axiom of cardinal extensionality. Thus R=FD=FUh_{(i)}=UFh_{(i)}. This demonstrates that UFh_{(i)} is a partitioning of R. □
Theorem 8.2
Proof
From the previous theorem we know that F induces a partitioning of R, by R=UFh_{(i)} (as F is onto). By definition cardD= cardR= cardFD= cardFUh_{(i)}=∑ cardFh_{(i)}≤∑ cardh_{(i)}= cardD, whence from cardFh_{(i)}≤ cardh_{(i)} it follows that cardh_{(i)}= cardFh_{(i)} as we cannot have equality if cardFh_{(j)}< cardh_{(j)} for any h_{(j)}. □
We also have:
Theorem 8.3
For everyFand every partitioning of the domainD=Uh_{(i)}it holds thatFh_{(i)}≠Fh_{(j)}⇒h_{(i)}≠h_{(j)}for arbitrary subsorts/subsets in the partition.
Proof
Assume the statement is false. Then for some F, and partitioning of the domain D: Uh_{(i)}=D, there are h′, h″ such that Fh′≠Fh″ and h′=h″. Suppose Fh′≠Fh″, while h′=h″. cardFh′∪Fh″= cardFh′+ cardFh″− cardFh′∩Fh″> cardFh′+ cardFh″ if cardFh′∩Fh″>0 or cardF[h′∩h″]>0. But F[h′∩h″]=0 only when h′∩h″=∅ and as h′=h″ this is not possible. Thus cardF[h′∪h″]= cardFh′∪Fh″≥ cardFh′+ cardFh″. But cardF[h′∪h″]=(h′=h″)= cardFh′⇒ cardh″<0 which is impossible. □
Theorem 8.4
ForFgiven as an arbitrary one-to-one functionD→RwithFD=R, cardD= cardR. For every partitionD=Uh_{(i)}and for anyh_{(i)}andh_{(j)}inDit always holds thath_{(i)}≠h_{(j)}⇔Fh_{(i)}≠Fh_{(j)}.
Theorem 8.5
For every onto collective functionFthere is at least one partitioning of the domainD=|H|=Uh_{(i)}, such that cardFh_{(i)}=1 for everyh_{(i)}in |H|. The (multi) set { cardh_{(i)}} is called the characteristic set ofF.
Theorem 8.6
Given a collective functionF, and characteristic partitioning of the domainD=Uh_{(i)}. It then holds for every objectsx_{1}andx_{2}inDthat: Fx_{1}=Fx_{2}⇔x_{1}andx_{2}both in somehinUh_{(i)}.
8.3 Collective Functions from Sort to Set
- 1.
not uniquely defined (up to permutation of objects between subsorts only)
- 2.when two objects are id:s themselves they can only be indexed for notational purpose. when h_{(1)}⊻h_{(2)}.
Theorem 8.7
(Cardinality Function)
- I
for every SetMwith cardM≤n, there exists at least one onto functionS→Mfulfilling the criteria of Sort→ Set functions.
- II
for any SetMwith cardM=k>nthere is no such onto function.
- III
this defines the same cardinal number as has been axiomatised for every Sort, choosing cardM=n. For any such onto functionFwhen cardM= cardSit holds thata in S, b in S, a≠b⇒Fa≠Fb.
- I
is just the existence Postulate 8.1 applied when the domain is a Sort and the codomain a set.
- II
is a direct consequence of condition (iii) in the definition of collective functions.
- III
For F one-to-one: cardD= cardFD=M=n by definition (n unique follows obviously from I and II).
8.4 Number of Functions Sort to Set
- (i)
A multiset of integers {n_{i}} representing a partitioning of S into subsorts h_{(i)} where cardh_{(i)}=n_{i} and \(\sum_{i=1}^{k} n_{i} = n\).
- (ii)The image {Fh_{(i)}}={y_{i}}, card{Fh_{(i)}}≤k where the integer n_{i} is assigned to y_{i}. Two functions will be different if either the multisets {n_{i}} differ or the sets {y_{i}} differ or both.From physics we recognize that the number of mappings in case b above,\(\binom{n-1}{k-1}\)is equivalent to the number of states in Bose-Einstein statistics.
- (a)
The number of into functions from S to M. Thus the problem is equivalent to finding the number of partitions of the integer n into k parts, which from combinatorics is known as being \(\binom{n + k - 1}{k - 1} = \binom{n + k - 1}{n}\).
- (b)
The number of functions from S onto M follows now from a) by demanding that card{Fh_{(i)}}=k i.e. every y_{i}∈M is the image of some h_{(i)}. This is seen by writing n=(n_{1}−1)+1+⋯+(n_{k}−1)+1=(n_{1}−1)+⋯+(n_{k}−1)+k and the problem becomes equivalent to a) by putting n→n−k which gives \(\binom{n-1}{k-1}\).
- (c)Consider now all possible functions from S to M. The largest possible k is obviously k= cardS=n representing the cardinality mapping of S. From (b) we will then sum over 1≤k≤n giving
- (d)
An injection, card(S)=card(FS), from S to M where k≥n will mean the conditions cardh_{(i)}=1, α≠β in S⇒Fα≠Fβ (where α and β represent any two id:s, not two in particular). This is easily seen to give \(\binom{k}{n}\) possible functions as each mapping defines a subset in M with card=n.
- (a)
8.5 Collective Mappings Set to Sort
A novel feature of possible mappings to Sorts is that the image of an element could be an unspecified (fundamentally unspecified) member of some subsort/subcollection of S,h⊆S. The notion for such unspecified element is the blur of the elements in \(h, \ \hat{U} h\). Given an element x in the domain (x∈M) we could assign the value \(\hat{U} h\) to x. In contrast to mapping Sort → Set the ambiguity is now found in the codomain with id:s as members. Again the definition of collective functions Sort → Set will be the approach. It is important to point out when dealing with Sorts, care must be taken when something is defined that the definition will not be inconsistent with the basic axioms regarding id:s.
Definition 8.4
- (i)
F(m_{1}∪m_{2})=Fm_{1}∪Fm_{2} where m_{1} and m_{2} are subsets of M.
- (ii)
Fx_{1}≠Fx_{2}→x_{1}≠x_{2} or as Fx_{i} in S, Fx_{1}≠Fx_{2} means Fx_{1}⊻Fx_{2}. As before the cardinality of the image is less than or equal to the cardinality of the domain.
- (iii)
cardFm≤ cardm.
- (iv)The image of a subset m⊆M defines in general a subsort in S with the usual meaning x∈m⇒Fx in h and for every α in h there is some y∈m such that Fy=α.
- (v)F can be specified by a partitioning of M into subsets M= ⋃^{k}m_{i} where for each m_{i}: Fm_{i} in S, FM=F[ ⋃^{k}m_{i}]=h in S, cardh=k. From this follows
- vi
Any permutation of the id:s in h leaves the collective function invariant.
8.6 Number of Distinct Functions Set to Sort
It was observed that the defining partitioning of the domain M= ⋃^{k}m_{i} is the only cause for distinction between two collective functions M→S. A partitioning of the set M is different from that of a Sort as we now have to have set-partitionings and not partitions of integers. As was seen each partition of M into k subsets will define an image of FM which defines a subsort of cardinality between 1 and k in S. Different subsorts will not give different functions if they have the same cardinality. Let us therefore determine the number of mappings for cardFM=k defining subsort h_{(k)} in S. This must then equal the number of set-partitions of M into k subsets. This number is known as a Stirling number of the second kind denoted S(n,k) with a recursive formula S(n,k)=S(n−1,k−1)+kS(n−1,k), k≤n. This gives the number of onto functions fromM→S. Next to get the number of into functions M→S is directly given from summing over all possible images each defined by cardFM= cardh=i giving \(\sum_{i=1}^{k} S(n, i)\). Finally to find the number of functions from a set M→S when cardM=n we will sum over all functions from M_{k}→S from k=1 to k=n, where M_{k} denotes any set with cardinality k, which sum is given by the Bell number \(B(n) = \sum_{k=1}^{n} S(n, i)\) recursivity given by \(B(n + 1) = \sum_{i=0}^{n} \binom{n}{i} B(i)\).
8.7 Sort to Sort Functions
9 Functions as Operators from and to a Sort
In the context of Sorts there is a clear difference between functions defined as classes of ordered pairs and as operations. Collective functions being some analogue to class of pairs were associated to sets of cardinal numbers. A function regarded as an operation on an argument to give a certain value can “naively” be seen as an operation of “exchanging something for something else”, assuming we can give some meaning to functions as operations defined when Sorts are in question. If two such functions have values in some Sort, they can never be distinct, for in that case there must be some x in the domain for which fx⊥gx which cannot happen when the range is a Sort. As there is no way to select a particular value in the range R, it seems as the only possibility is to say that the value fx in R is completely undetermined, which is equivalent to saying \(fx=\hat {U}R\) (the blur of R), whose value then can be seen as a “constant value” for f i.e. f a “constant function”.
9.1 Endomorphisms
- (1)
fx=x is a legal value as we have not changed anything, i.e. seeing f as an operator “exchanging something for something else”, fx=x represents “no exchange”. One can further impose the condition fx=x when x in h, h some subsort which accordingly is defined up to cardinality. This subsort then defines an invariant subsort relative f:s domain. Such an invariant subsort in f:s domain is then determined up to cardinality: fx=x when x in h, cardh=n say.
- (2)
Relating fx to some subsort h (if x in h does not involve a forbidden selection i.e. if we do not choose a particular value in h). Thus defining \(fx=\hat{U}h\) is an allowed value when x in h. The reason why we can consistently demand that x is in h is that x already is in the context of h and can thus relate to id:s in h, meaning that x cannot be confused with some x′ not in h. We could also “assign” the value \(\hat{U}Ch\), where Ch is the complement to h as x≠ all id:s in Ch. Continuing in similar ways to operators on two arguments fxy, one can define a limited number of different operators/functions. In this case one can create an analogue to binary operations. These operators can in turn be treated as new id:s forming Sorts of operators, etc. forming certain hierarchies of these.
10 Conclusion
In the second part of this paper I have been discussing consequences of introducing a weaker form of difference. This was called collective difference, characterized by lacking the normal individual difference when two objects are contrasted to each other, while being different in the sense of differently contributing to some common collective context, whole, collection etc. It was seen in the first part that the notion of order could be based on the basic notion of difference and similarity. To establish similarity between two things, reference to something third was seen as necessary. Therefore when regarding items with any collective difference this third item must be some collective common context expressed as: a is to C as b is to C, C representing some collective context. An example of this can be seen in the definition of a collective mapping F from a Sort: Fa=Fb where a and b are similar in relation to the common context of F which could be written as: a is to F as b is to F.
To form more mathematical structures working with Sorts, an important notion we have not been discussing at any length is operators, transformations etc. Due to non-ordinality such concepts will generate much poorer structures than in set theory, but might have aspects which are not naturally seen in a set-theoretical context. Certain classes of operators of Sorts representing endomorphisms could themselves be regarded as id:s, thus generating new Sorts. This procedure could then be repeated to yield hierarchies of Sorts, the structure of which might not easily be visible in a set-theoretic framework.
As a speculation, it is conceivable that not only particles could be indistinguishable in physics but also other aspects of a physical reality, such as physical states.
Notes
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