"We may define number psychologically as an archetype of order which has become conscious" (Jung).
"The number is in another space, not in this space" (George Spencer Brown).
"A set is a grouping into a whole of distinct and definite objects of our intuition or of our thought" (Cantor).
The Problem and its Approaches
The question of the nature of numbers and their relationship to sets has been the subject of debate throughout history. Is a number a set, a type, class, or category of set? Is the concept of number independent of the concept of set? Is number a primary concept or is it a derivative concept of set?
Gottlob Frege −the founder of modern logic, the greatest logician since Aristotle, the forerunner of the analytic philosophy of language and the philosophy of mathematics − published in 1884 "Die Grundlagen der Arithmetik" (The Foundations of Arithmetic), subtitled "A Logico-Mathematical Investigation into the Concept of Number", in which he set forth his logicist program, i.e., the attempt to base mathematics on logic. Arithmetic should follow the principles of logic and have no principles of its own. Since traditional logic was not sufficient for him to carry out this task, he was driven to create a new precise, flexible and powerful logic. This logicist philosophy, inaugurated by Frege, was later continued by Russell.
Thus, according to this philosophy, the concept of number had to be a logical concept. And since, according to Frege, the concept of set was a logical concept, he tried to ground the concept of number by means of the concept of set. If he succeeded, the concept of number would be analytic and a posteriori, a conception contrary to Kant's, who considered both arithmetic and geometry to be synthetic and a priori, since they are based on pure intuitions of space and time.
For Frege, logic tries to discover "the laws of truth," not the particular assertions of concrete truths. And since all sciences have truth as their goal, logic is the foundation of all sciences. For Frege, truths are indefinable and eternal; to find them one must go back to the logically simple.
Frege defined number from:
Property.
He defines property as a function that produces as a result a veritative value (true or false) for any object acting as an argument. "Every object either falls or does not fall under a property, the principle of the excluded third, tertium non datur" [Frege, 1998]. And he defines object as everything that is not a function. Property and object are logical concepts.
Extension of a property.
It is the set of objects that fall under that property. It is often referred to as a "class", but not in the sense of aggregate or collection. Examples: 1) the extension of the concept "cat" is the set of all cats; 2) the extension of the property "black" is the set of all black objects; 3) the extension of the property "trinity" is the set of all sets that have 3 elements. Frege was inspired by Bolzano's idea of a set: a group of elements that share a common property.
The property of equinumericity in sets. A property F is equinumerical with the property of G when objects of the extension of F can be paired one-to-one with each other biunivocally with objects of the extension of G. Sets having this logical property are considered equivalent.
From these 3 definitions, Frege defines a natural number as the extension of sets that have the same equinumericity property. For example, the number 3 is a set: the set of all sets of 3 elements: 3 horses, 3 umbrellas, and so on. Trieness is the common property of all such sets.
The number 0 is the extension of the property that the sets that are not equivalent to themselves have. The extension corresponds to the empty set.
1 is the extension corresponding to the property of being equivalent to the set containing 0.
2 is the extension corresponding to the property of being equivalent to the set containing 0 and 1.
3 is the extension corresponding to the property of being equivalent to the set containing 0, 1 and 2.
In general, a natural number is the extension corresponding to the property of being equivalent to the set formed by all the previous numbers.
For Frege, numbers are not attributes of physical things but of concepts. For example, if we say that the solar system has 9 planets, we are relating concepts. The number 9 is an attribute of the concept planet. "The assignment of number is an assertion about a concept."
Frege opposed empiricism, psychologism, and formalism:
Frege emphasized concepts. His axioms are conceptual, beginning with the concept of "property." He did not try to reduce arithmetic to a formal axiomatic system, for he was opposed to formalism, i.e., to considering arithmetic as an uninterpreted formal system and numbers as meaningless forms or objects. According to Frege, a formal system confuses the sign with the thing signified, which leads to the absurdity of identifying numbers with numerals, signs. For Frege, numerals are second-order concepts (concepts of concepts) that can also be defined in logical terms. For Frege there is always a meaning to every formal expression.
Frege also opposed psychologism, the idea that numbers (or mathematical entities in general) are subjective mental contents. Mathematics is not subordinate to psychology, but above it.
For Frege, logical entities are real, objective entities that dwell in a "Third World." It is a world beyond the physical and the mental, an abstract world, which is real, true, eternal, non-sensible and invisible. It is an archetypal world from which the physical and psychic world emerges. It is the world from which all laws emerge. "The laws of arithmetic are not laws of nature, but laws of the laws of nature, that is, fundamental principles about the thinkable." It is a kind of real, objective and eternal Platonic realm, where live mathematical entities that have their own existence, independent of the physical and mental world.
Frege rejected the position of philosophers such as J.S. Mill, for whom arithmetical truths are empirical truths, based on observation, mere abstractions of the properties of physical objects, and that arithmetic is founded on the induction of facts concerning groupings of physical objects.
In his later years, after his retirement in Jena in 1918, Frege admitted his complete failure in his attempt to clarify the nature of number, renouncing even his fundamental idea that numbers were sets. And thus abandoning his belief that arithmetic derived from logic.
His renunciation of the logicist program was replaced by his conviction that mathematics was properly geometry and that, therefore, arithmetic derived from geometry. He thus returned to the Kantian position he had fought against in "The Foundations of Arithmetic": numbers are based on intuition and arithmetic is synthetic and a priori. He thus established an analogy between numbers and geometrical objects. For Frege, number is analogous to a kind of geometrical entity or figure constructed on the basis of points grouped and organized in space. The point is the unit and the absence of points is zero. Forms, including numbers, emerge from geometry.
In this sense, Frege adopted a philosophy close to the Platonic one about geometry. Three sentences that illustrate Plato's philosophy of geometry are: "Geometry is the key to unravel the mysteries of the universe", "Geometry is the knowledge of the eternally existing", "Geometry existed before creation."
John von Neumann
John von Neumann −who made contributions to axiomatic set theory− simplified and formalized Frege's procedure. He defined the natural numbers directly as "pure" sets in the following recursive manner:
0 is the empty set:
0 = {} = ∅
1 is the set whose only element is the empty set, i.e., the one formed by 0:
1 = {0}
2 is the set formed by 0 and 1:
2 = {0, 1}
3 is the set formed by 0, 1 and 2:
3 = {0, 1, 2}
En general, n = {0,1,...,n−1}. A number is defined as the set formed by the previous numbers.
That is, every set is a set of other sets, and all sets are constructed from the empty set.
Peano
Peano's famous axioms of arithmetic, published in 1889 in "Arithmetices Principia, Nova Methodo Exposita", do not appeal to the notion of set:
1 is a natural number.
The successor of any number is a number.
1 is not the successor of any number.
No two numbers have the same successor.
Every property that 1 has and the successor of any number has all numbers. This is the axiom of induction.
John Conway
John Conway invented (or discovered) surreal numbers, a universal number system, because it can not only generate the natural numbers, but all kinds of numbers, including rational, irrational, infinitesimal, infinite, transfinite and hyperreal numbers, as well as operate with them.
As Frege and von Neumann did, he also defined them from the empty set. Conway's system is based on a set divided into two parts, symbolized by {x|y}, but formally it is a sequence of two sets. The number 0 is defined from the empty set as {∅|∅}, and from which all numbers are generated. [see Applications - Mathematics - Surreal Numbers].
The Solution: Numbers and Sets are Primary Independent Archetypes
As opposed to the logicist (Frege) and formalist (von Neumann, Peano and Conway) positions, in MENTAL number and set are different primary archetypes. They are two dimensions of deep reality that manifest themselves at the internal (mental) and external (physical) level.
Number is a primary archetype, as Jung and Pauli maintained. The set is also an archetype, which is different from that of number. Grouping (the result of which is a set) is not the same as counting (the result of which is a number). Sets are joined and numbers are added.
Of archetypes we can only see their manifestations. Every particular instance of an archetype connects us with that archetype.
Number and set, as archetypes of consciousness that they are, are inexpressible, that is, they cannot be explained (brought to the surface) because they belong to a deep level. Numbers and sets are linked to intuition.
Archetypes are a priori (prior to experience), profound, transcendent, intuitive, synthetic, invisible and inaccessible. Their concrete manifestations are a posteriori (linked to experience), superficial, immanent, analytical, visible and accessible, since they are in the objects of the real world. They thus unite the intuitionist and formalist positions. For example, no one has "seen" the number 3, for example. We can see the pure symbol that connects the superficial with the deep. The symbol of a number reifies the intuitive concept of that number.
Sets are not logical concepts. Logic is a dimension of reality that is based on a single primary archetype (Condition), and that manifests itself as deduction or decision, depending on whether or not it is accompanied by a generic expression, respectively. But there are more primary archetypes.
Frege's and von Neumann's definitions have the merit of constructing numbers from almost nothing, for the empty set is the closest thing to nothing in mathematics. The empty set (the set containing no elements) is not the same as nothingness, because its existence class is the same as that of any set. It is the only set that lacks elements and the only set that is a subset of any set.
This idea of a creation from nothing is natural in Hindu metaphysics, where all reality is considered to have been unfolded from a bindu, a point, i.e. a geometrical entity without extension, i.e. practically nothing. Nothingness is also identified with pure, unmanifested consciousness.
In MENTAL, nothingness is represented by the null metaexpression θ and is the content of the empty set:
({}↓ = θ).
Numbers are not sets. However, the sets constructed by Von Neumann from the empty set serve as primordial, paradigmatic, or representative sets of the cardinality of sets:
Numbers and sets are closely related. A set has cardinality (number of elements). And a sequence, besides also having cardinality, each of its components has an order number: from 1 to the number of elements. For example,
x=(a b)
x# // ev. 2
x# // ev. a
x#2 // ev. b
Numbers arise from sets when we abstract from their contents and consider their components as equivalent individualities. A number is a property of an abstract set, a form. But numbers are not sets, nor can they be expressed in terms of sets.
Number is such a powerful abstraction that it constitutes an independent entity, dimension or archetype. To this power contributed decisively the reification (representation) of numbers in the form of a finite set of symbols and positional notation, a recursive representation system that allows to represent any number by a sequence from 10 categories or classes of numbers (those represented by the digits from 0 to 9).
The number is an archetype and manifests itself in many forms. It can appear in pure form or as part of an expression as e.g. a sequence, a property, a quantity, etc. A quantity is a number of times a unit, where the unit can be any expression, e.g. 3*{a b c}, 3*apple, etc.
MENTAL contemplates abstract arithmetic (based on numbers) and concrete arithmetic based on quantities of objects, i.e.
(3*apple + 2*apple) // ev. 5*apple
(n1*apple + n2*apple) // ev. (n1+n2)*apple
Frege used an equivalence relation that makes all sets divide into equivalence classes. The numbers 0, 1, 2, etc. are the representatives of those classes. Each class corresponds to a number. When Frege speaks of extension he is really referring to an equivalence class. Each class n is the set of sets having n elements. Therefore, the numbers serve to classify the set of all sets.
Frege's Third World is the world of logical expressions, and would be equivalent to the world of MENTAL expressions, which are the manifestations of archetypes.
For Frege, mathematics and reality has a logical foundation. But the essence of reality (internal and external) is not exclusively logical; it is archetypal. Logic is only one dimension-archetype of reality.
Frege's "extension of a concept" is a simplification, for in reality there are fuzzy or subjective properties such as high, low, rich, poor, and so on. Therefore, the membership of the extension is also fuzzy. And a concept cannot be formalized as a function that takes only two values (T or F), but in general f*T where f is a real number between 0 and 1, with 0*T = F) and (1*T = T).
Peano's axioms in MENTAL
Peano's axioms constituted an attempt to formalize the natural numbers. But −as with any formal system− fails to capture the essence of the concept of number, and furthermore relies on the concept of "following" which it does not define and which admits many interpretations (spatial, temporal, etc.).
Peano's axioms expressed in MENTAL are:
1/num // 1 is a number
⟨( n/num → suc(n)/num )⟩ // the successor of a number is a number
( {⟨( n ← (n = suc(1)) )⟩} = ∅) // the 1 is not a successor to any nr.
⟨( (suc(n1) = suc(n2)) → n1=n2 )⟩ // no two numbers have the same successor
⟨( n/num ↔ n∈N )⟩ // if n is a number, it belongs to N
N = {⟨( n ← n/num )⟩} // N is the set of natural numbers
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