Gendai Shiso July 2020 Special Feature: The World of Sphere Theory — The Forefront of Modern Mathematics Reading Memo

Gendai Shiso July 2020 Special Feature: The World of Sphere Theory — The Forefront of Modern Mathematics Reading Memo

The theory of spheres is a mathematical theory that expresses concepts and structures in a general form and examines the relationships between them. A sphere here is an abstract mathematical object composed around two concepts, object and projection, and a sphere has the following three elements

• Object: The object that exists in the sphere and represents the abstract object. For example, a set or a topological space is an object.
• Morphism: expresses a relationship between objects, such as a “mapping” from one object to another. For example, a mapping between sets or a continuous mapping between topological spaces is a projection.
• Composition: It is an operation that combines two arrays and creates a new one. This composition operation must satisfy the coupling laws and there must be a constant projection.

A sphere can be composed of any kind of objects as long as they satisfy the above elements. For example, a set and a mapping, a topological space and a continuous mapping, or a graph and the projection between them can be considered as a sphere. Sphere theory allows one to investigate the properties and structure of spheres in a variety of fields, including algebra, geometry, computer science, and physics.

Also important in circle theory are concepts that express relationships between spheres. For example, a sphere isomorphism is defined when the structure of the object and the projection coincide between two spheres. Also important is a mapping between spheres, called a functor, which, like a circle homomorphism, has the property of preserving the structure between the spheres.

Sphere theory is related to set theory as described in “Outline of Set Theory and Reference Books“, logic as described in “Making Logic Part 1: Introduction to Logic“, algebra as described in “Structures, Algorithms and Functions“, formal linguistics and semantics as described in “Formal Language and Mathematical Logic“, and so on.

Sphere theory has applications not only in mathematics, but also in computer science. For example, the concept of sphere theory is used in the design of type systems for programming languages and query languages for databases. Furthermore, sphere theory has applications in various fields such as set theory, topology, algebra, and computer science, and is one of the most important theories of modern mathematics used in artificial intelligence tasks. In this issue, we present a reading note based on the book “Gendai Shiso, July 2020, Special Issue: The World of Sphere Theory — The Frontiers of Modern Mathematics,” which extensively discusses this theory.

A rich and fertile frontier of modern mathematics depicted by “arrows”.
Sphere theory is one of the most important fields in modern mathematics. Its high level of abstraction and generality is proving to be a powerful tool in many fields, including physics, computer science, biology, linguistics, and aesthetics. In this special issue, we introduce and discuss the basics of sphere theory, its various applications, and its philosophical implications, in order to get to the heart of this mathematical way of thinking that is attracting so much attention today.

Discussion
The rich interaction of thought opened up by the theory of spheres / Fumimoto Kato + Kouyato Saigo

Universality and Flexibility of Sphere Theory
Sphere theory is a relationalism between things.
The “sphere” is a category
A category is not a collection (list) of things
A category is not a list of things, but a means of comparing them.
Example
When we talk about the sphere of groups
It is not enough to just collect various groups
It becomes a “sphere” only when it includes the means of comparison, such as quasi-isomorphism, to compare their structures.
The relationship between the listed items is the protagonist
Universality
In general
There are individuals, and we think of universals as a set of them.
Universals” in sphere theory
It is an individual thing in itself
Universals are created when they are related to all other things in a certain way.
There can be multiple universals.
EGA of Grothandieck
Introduced sphere theory to algebraic geometry in earnest
Update the concept of space in algebraic geometry by using the word “sphere” and the word “shell
The concept of “scheme
Sekite, a kind of layer
Gelfand Transform
The ultimate generalization of the Fourier transform
A typical example of natural transformation in sphere theory
Sphere theory across boundaries – from AI to the theory of metaphor
Modeling the process of recognizing “sameness” in cognitive science using natural transformations
Applications of Sphere Theory in Cognitive Science
Research on systematics using the concept of association
Sphere theory has a strong affinity with graph theory
Commonly used in mathematical sciences related to networks
Higher-order spheres
Multi-layered networks within a network with further meta-networks
An approach to modeling backpropagation in machine learning by Sekite
Haven’t found a specific Yujin in the field of AI and machine learning
The idea itself is inspiring
Expectation that new ideas will emerge inspired by the idea of sphere theory
The framework of sphere theory will serve as a bridge, opening up the possibility of cross-disciplinary applications and developments in different fields.
Fusion of the sphere and probability
The sphere of “types” in the programming world
When something is expressed in terms of a sphere, the “meaning” of the thing is forgotten, and only the form of the relationship is preserved.
It is very syntax-like.
Don’t just look at programming as a sequence.
By collecting the semantics that are there, it is easier to find bugs where the meaning does not make sense.
The sphere seems to have only formality, and this may rather give rise to a perspective on “what is meaning?
Our world is filled with various “images”, and these images always have “associations”.
When you have an image of something
When there is an image, various associations run from it one after another.
It is possible to create a microcosmic sphere centered around an image.
The sum of these associations is “meaning.
Words are “sounds” and “letters”.
Mere sounds and letters have relatively few inherent associations.
When a network of associations is connected, it makes up a language.
Just as the addition of a subway line can significantly change the behavior of people in a region
Just as the addition of a subway line can change the behavior of people in a region, so too can the addition of a single cover-like line of association change the distance structure between various associations.
It would be nice to have a theory of meaning that could explain the fun of metaphors.
By using topos, we can
Natural language structures can be expressed in a typed lambda calculus.
In conjunction with these, we can model natural language processing in a sphere theory way.
Sphere theory is involved in reducing language to skin and bones
Towards a healthy abuse of sphere theory
When you’re trying to do something really creative
Sometimes you have to do a lot of childish, seemingly trivial, trials to get it right.
In physics, there used to be a popular saying about the “group plague.
Group theory was very effective in physics, especially in quantum mechanics
Physicists jumped on the “group theory” bandwagon, and the concept of groups overturned physics like a plague.
One of the most famous “abuses” of group theory is
Lévi-Strauss’s “basic structure of kinship”.
Support by mathematician André Veilleux
Klein’s very noble “abuse” of the quaternion group.
When people who are active in a certain field want to say something
When people in a certain field want to express something
They are impressed by the fact that they can express something in the framework of sphere theory, and try to formulate it.
They try to bridge multiple domains, such as mathematics and brain science, or cognitive science and programming, to find some common channel.
More inconvenient things happen than what is true.
Sphere theory is
not something very difficult like Goethe’s incompleteness theorem.
It is more fundamental and in a sense “lighter”.
Sphere theory has a “functional thing” called a projective as its main character.
It has a much more dynamic aspect than the earlier “set-driven” approach to mathematics.
However, it does not quite reach the dynamism of nature itself.
People who study cognitive science and consciousness complain about this.
While taking advantage of the universality and flexibility that already exists
Sphere theory itself could be made more mobile, which would open up a whole new range of applications.
Sphere theory is like a pizza crust.
What to put on it is important in applications. Translated with www.DeepL.com/Translator (free version)

Keynote/Introduction
Philosophy of Sphere Theory: From Spherological Structuralism to Spherological Unified Science / Yoshihiro Maruyama

1 What is Sphere Theory? From Sphere Theory as Mathematical Foundations to Sphere Theory as Scientific Foundations
Sphere theory was born in the middle of the 20th century in the development of algebraic topology.
It quickly established its position as a new “foundation of mathematics” to replace set theory.
Just as computer science was born from the theory of mathematical foundations (logic)
Sphere theory as a fundamental theory of mathematics was immediately applied to computer science.
It became the main methodology of theoretical computational geometry in the second half of the 20th century (Euro-type as opposed to the so-called American-type).
In this century, it has been applied to other sciences.
In particular, its application to physics has been remarkably successful.
Sphere theory is
essentially different from the holistic modeling of systems by differential equations
It gained a new status as a new kind of synthetic modeling language of science.
The scope of sphere theory science
Logic
Informatics
Physics
As a spin-off
Linguistics
Artificial intelligence
Economics and social choice theory
Sphere theory has moved in the last 50 years
From sphere theory as a mathematical foundation
Sphere theory as a new mathematical foundation” such as homotropic type theory based on high dimensional sphere theory has also emerged.
Sphere theory as a scientific fundamental theory has been transformed
There are even new startups with sphere-based technology at their core.
The sphere is
A sphere is a structural network of objects and projections (and composites of projections)
Ordinary graphs are also networks, but they have little or no structure
Purely abstract structure
Targets and projectiles are just placeholders
By filling in these placeholders, we can find a variety of health images in various areas of science
The three-way correspondence is called the Abramsky-Kooker correspondence
Logic has
There is a sphere of proposition and proof (deduction from proposition to proposition)
Proofs and programs are also deductive and computational processes.
Sphere theory has an aspect of general process theory.
Physics has
Physical science has the sphere of physical systems (systems) and physical processes.
Computer science has
There is a sphere of data types and programs.
In addition
Linguistics has
the sphere of words and semantic flow
In economics
There is a sphere of object and reasoned preference relations in economics, etc.
By formulating the various domains of science in terms of sphere theory
By putting the different domains of different sciences on a common ground of sphere
By putting different domains of different sciences on the common ground of the sphere, (de-centered) networks and transfers of knowledge between different disciplines become possible.
Example
By using the sphere theory correspondence between logic and physics
An automated inference system for physics can be constructed by using the correspondence between logic and physics.
Reintegration of knowledge through the development of spherological foundations across the various sciences
Reintegration of knowledge
Sphere theory as “modern natural philosophy
Each science has its own sphere
For any given sphere
For any given sphere, if there is a logical, physical, type theory to realize it
All spheres are logical, physical, and computational
There is a graphical calculus as a deductive system of intuitive “picture computation” associated with the sphere
In the good case, we can actually derive all of the equations of the sphere by pictorial computation, and
The completeness of the picture calculus can be proved.
A pictorial calculus mathematically justified by a strict definition of completeness
Sphere theory and its associated graphical calculus are
provide a pictorial computational language for the sciences that is both intuitive and rigorous.
Sphere theory is compared with set theory.
Set theory
If there is a thing, there is a collection, and then there is a collection of collections
By iterating this in a superlative recursive manner, a huge universe of sets is created.
What is essential to set theory
Superlative iteration
Set theory gives a picture of the world as a thing
Sphere theory of sets
Sphere theory is a vast generalization of set theory
What can be done in set theory can also be done in sphere theory
In sphere theory, things are just placeholders connected by arrows
Arrows are placeholders that just connect things
Arrows can be composed with each other in a vertical sequence (if the destination and destination types match)
The composition of these arrows needs to satisfy a simple property sail
If they are in the realm of things, they can also be composed in parallel
The idea that “arrows are primary and things are secondary” is false.
Each of the two types of existence exists in relation to the other.
Sphere theory gives a coterminous image of the world
In the sphere theory, the structure of the dichotomy between things and things is itself monetized, which in turn gives rise to things.
According to Johnston of Cambridge
According to Johnston of Cambridge, the 60’s was the age of monads (sphere theory universal algebra)
The 1970s was the age of the topos (sphere-theoretic universal geometry)
Then came the era of the monadic sphere
With logical counterparts
Linear logic and its reinforcement, sphere-theoretic quantum logic
In recent years, we have entered the era of higher dimensional spheres and higher dimensional topos.
Logic with logical counterpart
Homotropic logic
The “logic” of intuitionistic type theory
The “geometry” of homotropic theory
Logical framework to guarantee certainty and absoluteness of mathematics, and
Its computer implementation is crucial to the work of mathematics itself.
The Universality Axiom on the Identity of Existence (in addition to internal dependent type theory, etc.)
Based on
The Univariance Axiom
It defines the fundamental question, “What is the same?
It provides a formal framework for providing logical justification for mathematical arguments for the identity of equivalent objects.
The more traditional mathematical foundations of sphere theory
A place where “logic” and “space” come together.
The stage where intuitionistic higher-order logic and Grogan-Dandy algebraic geometry based on so theory merge
There are four kinds of sphere theory
Pure sphere theory as pure mathematics
There is no research community in Japan.
It is marginal in the world.
Applied sphere theory for pure mathematics
Sphere theory as a mathematical foundation
Sphere theory as a scientific foundation
Particular emphasis is placed on “compositionally”.
The origin of a system is regarded as a step-by-step composition of its original.
The system’s compositional origin is given causally.
Synthetic modeling is the main role of sphere theory in the sciences.
2 Sphere Theory Structuralism – Open Structuralism and Structural Realism
Mathematics is the art of giving the same name to different things.
The art of mathematics is to unify variety by finding structural equivalence between various different entities.
Sphere theory is not a reworking of traditional structuralism.
It is a new kind of structuralism.
Lévi-Strauss used the Bourbaki structuralism of Veilleux and others as a backbone.
The structure in Bourbaki structuralism is
The structure in Bourbaki structuralism is first of all a plain set of entities as the original basis
On top of that, we can add algebra, topology, order, and other additional structures as toppings.
Set-theoretic structuralism is certainly convenient for increasing the number of structures step by step.
But
However, as long as the entity of set exists as a “given myth” in the beginning, it is not a theory that directly captures the idea of structure.
For Bourbaki, “structure” is the language in which mathematics is practiced.
The same can be said about model-based structuralism.
Unlike the undefined notion of “structure” in ordinary mathematics
“structure” is a technical term in model theory
Structure, or in its special case model, is treated as a mathematical object in itself.
“Structure” or its special case “model” is treated as a mathematical object in itself.
In model theory, “structure” and “model” are
after all
Bourbakian formulations of structural concepts in predicate logic.
Model-theoretic structuralism is
sensitive to the difference in the concept of “structure” between algebraic and topological structures.
In the above sense
It can be said that it is a more precise structuralism that solves the subtleties of the “structure” concept than Bourbaki structuralism.
Ultimately
Ultimately, it is nothing more than a version of set-theoretic structuralism that posits a “given myth” of entities in the beginning.
“Structure” is usually an undefined term in mathematics.
There is no clear definition of what structure is in general, apart from individual cases
Among mathematicians
Among mathematicians, the concept of “structure” is shared and there is a certain common understanding.
In the same way that naive set theory defines the cardinality of a set as an equivalence class
In the same way that the radix of a set is defined as an equivalence class in naive set theory, it is not impossible to define “structure” as an equivalence class of instances of a structure.
In that case, too, the concept of “structure” can be defined only if the entity of the instance of the structure exists first.
Even “algebraic structures” and “topological structures” do not have a clear definition of structure.
It is not obvious whether a group as an algebraic structure is a group as a syntactic algebraic theory or a group as a semantic model-theoretic structure.
Sphere theory is structuralism without a “base set
3 Sphere epistemology and sphere ontology – absolute, general, and conceptual foundations of knowledge and existence
Classification of academic foundational theories
Absolute foundationalism
The kind in which all existence and knowledge of the discipline are reduced to a single framework.
Relative foundations
The kind of foundations in which different ontological and epistemological frameworks are used depending on the kind of being and knowledge.
Local or structural foundations
Conceptual foundations
The kind of foundational theory that is concerned with the pursuit of concepts that play a fundamental role in the practice of the discipline.
Epistemological significance of sphere theory
Higher-order lawfulness” as opposed to first-order lawfulness
Structuralism about openness is higher-order structuralism
Explicit formulation of “analogy between analogies
Words by Banach
Abramsky-Kutka correspondence describing the trinity of logic, physics, and computation
On higher-order meta-lawfulness across the three domains
The ontological significance of sphere theory
In its intrinsic multiversality and multiplicity, and in the principle of lightening ontology by structural ontology
Sphere theory diminishes ontological commitment by its structural ontology
Lightweight ontology is achieved by avoiding commitment to real entities and committing only to entities as structures.
The question arises as to why the subjective existence of existence in our minds can have objectivity and why the truth about it can have absoluteness.
An easy ontology makes epistemology difficult, and an easy epistemology makes ontology difficult.
Structural ontology involves
Structural ontology has the solid ontological advantage of maintaining a realism about structure and its status as a realism, while at the same time solving the problem of cognitive access to that reality.
4 Spherological Unified Science – Vienna School, Stanford School, Oxford School
Stanford School emphasizes the pluralism of science
Urged revival of the meaning and strength of disparate
On the other hand, the Vienna School is a monistic, reductionist, and foundationalist unified science.
In general, the various sciences have different epistemologies and ontologies.
It is precisely because they are different that collaboration is meaningful.
Unification that leads to uniformity is rather harmful.
The most powerful substance was impure substance
The hybrid nature of science creates strength and stability of science
Adopting a position of scientific pluralism does not mean giving up on unification and integration
What the Stanford school denies is
The Stanford school denies the possibility of a reductionist unification between first-order laws at best.
The Vienna School of Unification Science is, so to speak, “unificationism from above.
Sphere theory allows for
Network unification of higher-order laws
Sphere theory unification science is “unification science from below
Sphere theory as a mathematical foundational theory has been indexed by Roveala
Sphere Theory as Absolute Meaningful Foundational Theory
Sphere theory as a scientific foundational theory has been well driven by Abramsky and others
Sphere theory as a relative foundational theory Translated with www.DeepL.com/Translator (free version)

What the Sphere Was Like / Mariko Kohara

Introduction
What is the sphere?
The theory of the sphere is, simply put, “a grouping that also considers relationships.
Example
A family consists of four members.
Considering the relationships within the family
Similarities between nuclear families can be compared
Undergraduates and Spheres and Sets
Sphere and Kante in Algebra
What is a sphere?
Category
Data called object A,B,C,… and
and a class of projections (morphism or arrow).
If we take a projectile f from the class of projectiles
For f
There is a target dom(f), called the domain of f, and
and the object cod(f), called the codomain of f, are determined.
Let dom(f)=A and cod(f)=B
Write f as f:A→B
Certain “conditions” are satisfied
For any projections f and g such that cod(f)=dom(g)
We can define a projective g∘f:dom(f)→cod(g) which is called a composition of f and g.
They satisfy the following two properties
Combinatorics
For any f:A→B, g:B→C, h:C→D
satisfy h∘(g∘f)=(h∘g)∘f
Meaning that the parenthesized part is an equality even if it is calculated first
Existence of inequality projections
For any object A, there exists a projective 1A:A→A, and
For any projectile f:Z→A
1A∘f=f
or for any projective g:A→B
g∘1A=g
For each projectile, there are two objects at the beginning and end of the projectile.
What is a class of projectiles?
Roughly speaking, it is a collection of projectiles
What is the difference between a sphere and a set?
A set is a collection of things
Each thing that makes up a set is called a source (or element) of the set
The following is a concrete way to give a set X consisting of a collection of “things with the property R”.
Mapping between sets
For a set X and Y, for any source x of X, there is a correspondence f that defines exactly one source of Y, called f(x).
X is called the defining region of Y
The set {y|y is a source of Y and y=f(x)} for some source x of X is called the value range of f.
Y is called the end set of f
When Y is a set of real numbers, f is called a function.
What is the difference between the notion of object and projection in the sphere and that of the whole set and the projection of a set?
In the sphere, unlike the mapping of sets, the associative law is required from the beginning.
In sets, homogeneous maps can be constructed naively
In the sphere, the existence of isometries is defined in the first place.
A “set consisting of all sets” is not a set
In the theory of the sphere
In the theory of spheres, we can create a “sphere of sets” by treating each object as a set and the projections as projections between sets.
A “set consisting of all sets” is not a set.
A set is a collection of “things.
What is a “thing”?
Assume that a set consisting of all sets is a set.
The set consisting of all sets as a whole is further divided into two sets
One is the set Y, which is the set of all sets that have themselves as their source.
The second is the set Z, which does not have itself as its source.
Z itself is not the source of Z
Z is not the source of Y either
We just divide all the sets into two, and suddenly we have a set Z that belongs to neither of them
Irrationality
Contradictions arise because the discussion is carried on with the ambiguity of what a set is.
Set theory that does not obscure them
A set theory in which the properties of possible sets are observed in detail and defined as axioms.
Axiomatic systems are
Axiomatic systems are written using “logical formulas” that have been developed to describe “objective conditions.
I want to state an objectively correct condition.
You need a logical formula and a set of rules for inference.
A set is said to be a set if it satisfies the above retail
ZFC Axiom System
These axioms plus the axiom of choice
Under this axiomatic system
Under this axiomatic system, mathematics such as calculus and linear algebra are performed and given a usage name.
Considered to be sufficient to describe modern mathematics
The set of all sets that appear in Russel’s axiom
The set of all sets that appear in Russel’s inverse is not a set because it cannot be constructed by this axiomatic system.
against axiomatic set theory.
In contrast to axiomatic set theory, “collection of things” is also called naive set theory.
What is a “class” in the definition of a sphere?
In axiomatic set theory, a collection of sets is not necessarily a set.
In axiomatic set theory, a collection of sets is not necessarily a set, so the collection of sets is called a class.
Under the ZFC axiomatic system, a class itself cannot be handled inside a body type.
A class is formally treated as a “logical formula with variables”.
Sphere of a sphere
If the object of a sphere is a sphere, what does the projection do?
From sphere to sphere, there is a correspondence called a Kante
If C and C’ are spheres
If F is a covariant move from C to C’ (say F:C→C’), then
For an object X of C, there is only one fixed object of C’ (let it be F(X))
Then, for every projection f:A→B of C
Then, for every projection f:A→B of C, there is a projection F(f):F(A)→F(B) of C’.
For this correspondence
F(f∘g)=F(f)∘F(g).
For the above correspondence
For every projective f : A → B of C, there is a projective F(f) : F(B) → F(A) of C’.
For this correspondence, F(f∘g)=F(g)∘F(f) is called
is called an invariant function.
And the covariant operator from sphere C to C
Transfer the object X of C to X
The mapping of a projective f to f is called
is called the constant operator of C and is written as “1C”.
If you want to compare different spheres
If you want to compare different spheres, you can do so by using the appropriate operator
For the spheres C and C’
There exist covariant functions F:C→C’ and G:C’→C
The composite F∘G:C’→C’ and G∘F:C→C are
F∘G=1C and G∘F=1C, then
C is a sphere isomorphism to C’
If F∘G and G∘F are isomorphic to the homotopy 1C and 1C respectively, then
C is called sphere isomorphic to C’.
What is a homomorphism as a Kante?
Define a surjection (natural transformation) between the Kante
Let L and H be the covariant operators from sphere D to sphere D’.
What does it mean that U is a projection from L to H?
For any object X in D
The projection u(X):L(X)→H(X) is fixed and
Figure 1 is commutative for any projection h:X→Y of D.
It is an isomorphism of Kante L and H as Kante.
There are projections u : L → H and v : H → L of the Kante.
For any object X in D
u(X)∘v(X)=1H(X)
v(X)∘u(X)=1L(X)
Equal as a joint move
Isomorphic as an artefact
Looser than above
Conditions are stated using the projections between the Kante and the Kante
Basic definitions of things in sphere theory
A sphere consists of an object and a projectile, and there is a correspondence from sphere to sphere called a Kante.
Using Kante, we can compare spheres with each other.
The problem of what to do with the sphere with me as an undergraduate student
After defining the sphere, what do we use it for?
Prove something to yourself after rewriting a story in the framework of the sphere
Most people can’t stand abstractions without a purpose
Graduate students and the various spheres as tools
There are three patterns in sphere theory
Treat set theory and sphere theory as logic
Using the framework defined in sphere theory to describe other pure mathematics in an understandable way
Integer Theory
The field that investigates the properties and behavior of prime numbers
Review sphere theory in terms of practical applications such as programming
Examine the quantities associated with the sphere
Consider the sphere as an elementary school student
Assume that there is only a simple homotopy
Consider the spheres of junior high school and high school students
If we calculate the percentage of people who watch evening cartoons, each sphere will have one new quantity associated with it.
Similarly, if we “calculate the percentage of people who watch late night anime” in each sphere, we will get one new quantity for each sphere.
The “quantity associated with a sphere” is called “quantity” whether it is a number or a set.
By comparing these quantities, we can determine the nature of the sphere.
Algebraic K-theory
The quantity associated with a sphere that satisfies the appropriate conditions
If we take one sphere that satisfies the conditions, we can determine one quantity, which is called algebraic K-theory.
In integer theory
An important quantity that reflects the properties of integers and primes.
What properties of the sphere does algebraic K-theory, as defined in general, reflect?
It was not well understood what properties of the sphere made it an important quantity.
A quantum leap forward for algebraic K-theory
The sphere called the higher-order sphere
Using the theory of higher-order spheres
Algebraic K-theory is a quantity that satisfies two properties of the appropriate sphere
On the other hand, the quantity satisfying the two properties of a proper sphere can be determined only by algebraic K-theory.

About the Higher Sphere
Example
Vectors
A vector has two data, direction and magnitude, and can be represented by a set of components consisting of real numbers.
Example
Coordinates in the xy-plane correspond to quadratic reality vectors
Vectors with the same number of components can be considered as an addition
We can also consider multiplying by a real number
Real vectors with the same number of components form a set
Addition and multiplication of real vectors are also included in the set again
This is called real vector space
In particular, a vector space consisting of vectors with n components is called an n-dimensional vector space
A mapping from one real vector space to another real vector space
Take a real vector as the destination real vector
The destination real vector of the standard vector in the linear mapping is called
If we write the destination real vector as a linear combination of the standard vectors in the destination real vector space, we get a matrix representation
Example
Suppose the linear transformation f maps the standard vector (1,0,0) to a11(1,)+a12(0,1), the standard vector (0,1,0) to a21(1,0)+a22(0,1), and the standard vector (0,0,1) to a31(1,0)+a32(0,1).
The destination of (x,y,z) by the linear map f can be calculated as the product of the matrix and the real vector
If we keep one basis, a matrix corresponds to one linear map
Addition and multiplication of matrices is also a matrix.
The entire set of linear maps of M-dimensional real vectors from N-dimensional real vector space is
becomes a set closed to addition and real multiplication
In other words
The entire linear map of a real vector space is again a real vector space
Consider a sphere with vector spaces as objects and linear maps between them as projections
Each object is an N-dimensional real vector space or M-dimensional real vector space
Each projection is a linear map
The sphere of a real vector space is a real vector space in which the class of projections between two objects becomes a real vector space again.
The sphere of a real vector space has a rich structure that cannot be described only by the existence of associative laws and homogeneous projections.
When we consider the class of projections between any two objects
When we consider a class of projections between any two objects, we call it a “fecundity sphere” if it has an additional image to eat.
The sphere of a real vector space is an example of a fertility sphere.
A sphere in which the class between any two objects is a set
A fertility sphere is a sphere in which the class of projections between any two objects is a set, such as the sphere of a real vector space.
The set in which the notion of “neighborhood” is defined is called a “topological space”.
ε-delta argument
If a sequence of numbers converges to a real number α
We want a sequence of numbers with a sufficiently large display to fit into an open interval around Α whose half-width length is ε.
This open interval whose length is ε in half its width is
This gives us
The set of real numbers is a topological space
Using an ε-neighborhood
The continuous function on the real numbers is
A continuous function on the real numbers is defined to be a function with the property that if x is brought closer to α, f(x) will fall within an “ε-neighborhood” of f(a).
Similarly, in the phase space
Similarly, in phase space, we can define a continuous mapping using the neighborhood.
We can form a sphere with the phase space as an object and the projective as a continuous map.
If we add the condition “compact generating space” to the topological space
If we add the condition “compact generating space” to the topological space, the whole “lotus” map from one compact generating space to another becomes a compact generating space again.
The sphere of a compact generating space is
the sphere formed by the compact generating space becomes the sphere generated by the compact generating space.
Another example of fertility sphere
Unit
An object consisting of vertices, edges, faces or oriented figures of larger dimension
What can be imagined from the above
A polyhedron with arrows on the edges
A tetrahedron whose edges are arrows and whose faces also have arrows
What is N alone?
A set of numbers from 0 to N, ordered by size.
A figure in which each number from 0 to N represents a vertex.
Between the abstractly defined N-units and M-units
between abstractly defined N-units and M-units, we can consider a map of sets that preserve order with respect to size.
The sphere called “simplex category” is formed from these abstractly defined objects and the mapping that preserves the order.
An antivariant move from this simplex category to a set is called a simplex set.
If there is one antivariant Kante, there is one unitary set.
Since a unitary set is a Kante
We can think of projections (natural transformations) between the Kante we defined earlier.
Since a unitary has a geometric interpretation
We can give a tree-price interpretation to a unitary set
Example
Create an oriented figure for each N by connecting the origin of an N-dimensional real vector space with the remaining N heavens.
There is also a unitary set that can be interpreted as if N is moved from 0 to infinity and the figures are stacked on top of each other.
A unitary set is like a topological space
The sphere generated by a unitary set is called a simplicial category.
Since a unitary set has the same properties as a topological space
We can consider “associated quantities” such as homology groups and homotopy groups.
For two unitary sets, if their homotopy groups are isomorphic
Consideration of the unitary sphere by Dwyer and Kan
Diagram of the part of the set of projections from object X to object Y in the unitary sphere (= unitary set) that is the image of k units.
The set of projections between two objects becomes very simple when a certain condition, called the calculus of fractions, is attached to the sphere.
Me as a graduate student and applications of sphere theory
Postdocs and sphere theory in various fields
Homotopy Theory and Algebraic Geometry
Real vectors had real doubles defined
Real numbers have addition and multiplication
Addition and multiplication of real vectors were obtained by using addition and multiplication of real numbers for each real component
Even if it is not a set of real numbers
Even if the set is not made up of real numbers, if the set is such that addition and multiplication are defined
If the set is not a set of real numbers, but is such that addition and multiplication are defined, then a vector can be defined with the source of the set as a component.
A set that provides addition and multiplication where the answer remains the same even if the number to be multiplied and the number to be written are interchanged
We can consider vectors whose components are the elements of a commutative ring, create vectors from a commutative ring, and consider the sphere of a vector space created from a commutative ring.
What is algebraic geometry?
To study commutative rings and vector spaces created from commutative rings
To study the quantities associated with the sphere of a vector space created from a commutative ring, called commohology or homology.
More generally, it is the study of various properties of topological spaces formed from commutative rings. Translated with www.DeepL.com/Translator (free version)

Mathematica/Logic
Sphere Theory and Topology / Dai Tamaki

1Origin of Sphere Theory
The notions of sphere, Kante, and natural transformation were first introduced by
The notions of sphere, Kante, and natural transformation were introduced by Eilenberg and MacLane in order to briefly describe the properties of homology.
What is homology?
It was introduced by Poincaré to study what he called manifolds.
What is a manifold?
A geometric structure defined as a generalization of a curve or surface.
Examples of manifolds
One-dimensional manifolds
Two-dimensional manifolds1
Exercises can be drawn on a spherical surface, but they can be continuously copied from one another no matter how they are drawn.
Imagine a rubber band placed on a ball and moved by sliding it over the ball.
Such an exercise can be continuously reduced in size to a single point.
b1(S2)=0 since there are no intrinsic one-dimensional cycles on the sphere
Two-dimensional manifold 2
When you put a rubber band on a torus, if it is missing in the “vertical” direction, no matter how hard you try, you can’t crush it into a single point.
Although it is difficult to apply a rubber band from the outside in the horizontal direction
But if you think of the torus as a float, you can put a rubber band on the inside of the float in a horizontal direction.
The rubber band cannot be crushed at a single point.
Nor can you fix it “vertically” by continuous deformation.
On the torus there are
There are two essentially different 1D cycles on the torus
The one-dimensional Betti number of the torus is 2.
In general, the notion of an n-dimensional cycle is defined as
The idea of “investigating the properties of a manifold by its essentially independent number” was introduced by Riemann and Boetsch
Example: Betti numbers for spheres and torus
We can prove that the sphere and the torus are different
Poincaré’s approach to identifying properties of manifolds
Arbel group
Set with additive + and subtractive – defined
The notion of category is defined by abstracting from the properties of sets and mappings
An object corresponds to a set
The object corresponding to a mapping is a morphism
The sphere Set, whose object is a title and whose map is an operation
The sphere Abel over the Abelian group and quasi-isomorphism
The sphere Vect over a vector space and a linear map
Top over topological spaces and continuous mappings
2 Homotopy and higher-order spheres
The theory of spheres was born in this way.
This theory of the sphere has been used in various fields since then.
Grothandieck was the first non-topologist to use sphere theory in earnest.
He reconstructed algebraic geometry from the ground up.
Furthermore
Sphere theory can be viewed as an algebraic and geometric object
Basic ideas in viewing the sphere as a geometric object
The relation between the sphere and the triangle
Two composable projectiles form a triangle
Two alone
From three composite projectiles, we can form a tetrahedron in the same way
3 single units
A prefecture can be regarded as an object made up of units of various dimensions.
A figure that is defined as a collection of units is called a
A figure defined as a collection of units is called a unitary complex.
To consider the sphere as a geometric object is
This view of the sphere as a geometric object is closely related to the current development of the theory of higher-order spheres.
In higher-order spheres, it is necessary to express relations between various kinds of projectiles.
This alone can be very complicated.
In such a case, a geometric interpretation using single objects is very useful

Number Theory Geometry and Sphere Theory / Satoshi Ito

1The “Idea” of Number Theory Geometry and the Theory of the Sphere
In the field of mathematics, there is a field called “number theory” or “number theory”.
In short, it is a field that studies the “laws of numbers.
Fermat’s Last Theorem
There is no positive integer that satisfies
The field that studies the laws of numbers using geometric methods is called number theory geometry.
The Idea of Number Theory Geometry
There are many laws of numbers that have not yet been clarified. There are many laws of numbers that have not yet been clarified, and some of these laws can be clarified by applying “continuous” methods such as topology and physics.
Slogan of Number Theory Geometry⁉
In order to further develop number theory geometry, it is important to incorporate the methods of topology and physics into number theory geometry using sphere theory.
2 The path paved by Gauss
At the age of 15, Gauss was interested in the growth of the prime number sequence 2, 3, 5, 7, 11, … and conducted an experiment.
Let π(n) denote the number of primes less than or equal to n. Since there are four primes less than or equal to 10, π(10) = 4.
Then, it was experimentally found that π(n) increases slowly as n increases.
Prime Number Theorem
The elaboration of the prime number theorem is related to the Riemann prediction.
Mutual Law of Square Surplus
Let p and q be odd prime numbers that are different from each other.
1. if p or q minus 1 is divisible by 4, then it is the same ground that p is square remainder with q as law and q is square remainder with p as law
2. otherwise, the necessary and sufficient condition for p to be square surplus with q as its law is
otherwise the necessary and sufficient condition for q to be square surplus with p as its law is that q is not square surplus with p as its law
3. Galois theory of Grothandik
4Mutual laws and conservation of square surplus
5Veile conjecture
6Etale cohomology
7Ramanujan Conjecture
Ramanujan Conjecture
8On the proof of the Veilleux conjecture
9Homomorphism conjecture, Langlands Kantianity, and “sphere theorizing” of homomorphic representations
10Basic complement, Hitchin-fibration, and mirror symmetry
11The sphere of mixed motifs
12 Conclusion

A Guide to Sphere Logic — A Topos Connecting Logic and Mathematics / Nagafumi Aratake

1 Introduction
Sphere Logic is
Mathematical logic is the study of the interaction between sphere theory and logic by representing objects in mathematical logic in terms of sphere theory.
to study the interaction between sphere theory and logic.
Three paradigms are important in sphere logic
Kantian semantics
Sphere theoretic interpretation of logical formulae
Correspondence between theory and sphere by internal language of formulae
2The Beginning of Sphere Logic–Rovere’s Kantian Semantics
Kanteki semantics for algebraic theory
To introduce the idea of Kantian semantics
Start with “Sphere Set of Sets and Mappings
3Sphere theoretic interpretation of more complex logical formulas, and models in the “logical sphere
4From Logic to the Sphere and from the Sphere to Logic
Introduction
Interludes
5Topos Theory: A Bridge between Logic and Mathematics
6Conclusion

Sphere Theory and Set Theory / Masashi Fuchino

1Foundations of Mathematics and Mathematical Foundations
Foundations of mathematics are the basic knowledge and concepts necessary for studying, researching, and applying mathematics.
Foundations of mathematics” refers to the study of the foundations on which mathematics is built.
A framework for mathematically analyzing our actions of doing mathematics from the outside (meta-mathematics)
Goethe’s Incompleteness Theorem
The difference between set theory and category theory
2Set theory as a mathematical theoryforeverything
Set theory can be described as a subsystem of all existing mathematical theories.
Most of the treatment theories in mathematics are just encoded as subtheories of set theory.
3 Small categories and large categories
Review of the definition of a category
What is a category 𝖪?
K is the range of the object and
The range of an object is not a set in many cases, but a true class
In the standard axiomatic system of set theory, ZFC (Zemelo-Frankel set theory with selection axioms), classes cannot be treated as objects of the theory.
Each concrete class can only be treated as a separate chapter of the logical formula that defines it.
Example
A class consisting of a whole group
We will give this class as a separate chapter of the logical formula Φ(x), which corresponds to “x is a group” in the language of set theory
obtained by specifying a set Hom(X,Y) of what are called morphisms between any two objects X,Y in its range.
The definition of a category is made as a known notion of a set.
For any object X, Y, Z in K and for each element f, g of Hom(X, Y), Hom(Y, Z)
Their composition f∘g∈Hom(X,Z) is associated with it.
4 Grothandic Universe
For the ZFC axioms, we make them available to sphere theory by adding axioms.
5Levi-Montague reflection theorem
6Vopenka Principle and Categories
7Set-theoretic multiverse as the ultimate category theory

Computing/Language
Recollections and Thoughts on Computer Science and Sphere Theory / Hiroyuki Miyoshi

1Algebraic Semantics and Goguen
The Connection between Computer Science and Sphere Theory
S.Eilenberg used sphere theory to develop his theory of automata
The Mathematics of Automata Theory
Algebraic semantics of M.A.Arbib & E.G.Manes
A series of works by J.Goguen
Construction of primitive algebraic semantics by ADJ group
Algebraic semantics or algebraic data types
The reason why Goguen does not get into complex sphere theory, but still uses it, is that it is a framework for him to find a balance.
For him, sphere theory was a framework to achieve balance.
Computer science is a field that requires a balance between logical, physical, and human aspects
Ease of understanding for humans is an important factor
Curry-Howard-Lambek correspondence
Cartesian closed sphere and simple type theory, higher-order intuitionistic logic and elementary topos
A period of rapid exchange between mathematical logic, computer science, and sphere theory
In 1986
Logic in Computer Science (LICS) was started in 1986
Applied to Coquand-Huet’s Calculus of Construction (CC), a powerful impredicative type theory
Designed and developed theorem verifiers such as LEGO and Coq
Theorem verification requires a type theory with hierarchy
Goguen’s primitive algebraic semantics is not too difficult for experts in sphere theory
2 Representational Semantics and Computational Monads
In elaborate representational semantics
Giving meaning to programs
In a real programming language, it is important to express side effects and exception handling, including input/output.
Therefore, the translation of syntax into semantic domain became more and more complicated
Even if you give the semantics of a program, its translation becomes as complex as the original program
This is why operational semantics became popular, as it is intuitive and easy to understand.
E.Moggi came up with the idea of using the monadic structure for discussing algebra in terms of spheres as a framework for organizing and representing representational semantics.
In functional programming languages
When considering side effects
Include side effects in the output of the function
The problem is function composition.
Because side effects are included in the celebratory power
When we include side effects in the celebratory power, we lose the ability to synthesize functions that were originally synthesizable.
The solution to this problem is
The solution to this problem is the monadic structure projections (natural transformations) µ and 𝛈.
By using these, we can systematically redefine the new composition.
The function considered in this way is called the monadic function
It is also compatible with Curry-Howard-Lambek compatible shoulder theory
By interpreting monads (and commonads) as aspect operators in logic
By interpreting monads (and comonads) as phase operators in logic, we can consider extensions of Curry-Howard-Lambek correspondence.
Haskel, but not a representative language that uses monads
It’s a pure functional language, with side effects, especially IO.
The use of computational monads has had a profound effect on the design of the language itself.
By assuming that the monad hides computations such as side effects
The monad can be seen as a kind of computational reflection.
It has come to be used as a theoretical framework for considering programming languages with two levels of implementation: the hidden level and the explicit level.
Recently, the algebraic effect has been considered as an alternative to monadic computation.
It can capture various side effects more intuitively than computational monads.
Easier for humans to understand.
3Applied Sphere Theory and Limitations of Sphere Theory
Consideration of applications to various fields by the sphere theory community
J.Lambek’s sphere-theoretic calculus and multisphere corresponding to substructural logic with applications to linguistics in mind
Pioneering work in linear logic by
Application of sphere theory to the description of hierarchies in organisms by A.C. Ehresman
Application of sphere theory to cognitive science
S.Abramsky and B.Coecke’s work on sphere-based quantum theory
Finite free any quantum system treated in quantum information.
by using various compositions of string diagrams in sphere theory.
characterize quantum phenomena such as quantum teleportation using sphere theory.
Research on graphical spheres
The relationship between knots and sphere theory through braid groups
Study of its abstraction, the Joya-Street tensor sphere
Graphical formulation of the Geometry of Interaction by Abramsky-Jagadesen
Hyland-Street-Verity’s monoidal sphere with trace
Applications of Lambda Calculus to Semantics, found independently by M. Hasegawa and M. Hyland
Survey by Selinger
A qualitative understanding of Abramsky-Coecke’s quantum theory in finite dimensions
Interactions, two-way information flow, and the property of composability are applied outside of quantum theory.
The MIT group led by D.I. Spivak in the U.S. and the flow of applied area theory led by J. Baez
Organization of an international research conference (Applied Category Theory)
Creation of the journal Compositionality
Description of the Fong-Spivak-Tuyeras backpropagation
The direction of interesting applied research rather than difficult basic research
Many technical topics are not so difficult mathematically, which can be done by talking about strict spheres
Sphere theory alone does not easily lead to applications
Examples
Computational Reflection
Raised in the context of programming languages
Reflection itself is
Nowadays, in advanced programming languages
metaprogramming and generic programming.
I will discuss reflection in the context of Smith’s paper.
Using Lisp
Preparation
Think of a program and an executor as an abstract program and its execution model.
The executor is itself realized in some programming language and its executor, such as an interpreter.
The above assumptions can be traced back infinitely to level 3, 4, 5… and so on indefinitely.
When the language used to describe the executor at each level is the same
Smith assumed that this hypothetical infinite hierarchy is in operation when the program is executed, and called it 2-Lisp.
The behavior, or semantics, of the program
It doesn’t make any difference whether you assume this kind of refurbishment or not.
As program behavior
You can see the information of the behavior of the upper level, and
Modification
What happens if the program can see and modify information about the behavior of the level above it?
The above would make it possible to change the behavior without changing the program.
The addition of such instructions to the 2-Lisp language was called 3-Lisp.
Smith called this self-representation in the sense that the things that can be reifyed or reflected define the range in which one’s own behavior can be modified.
In fact, in 2-Lisp, the only things that the executor can manipulate are expressions, environment, and continuation.
In fact, in 2-Lisp, the executor can manipulate only expressions, environments, and continuations, while in 3-Lisp, it can manipulate these at the next level up.
Self-rewriting becomes possible in that range.
Example
The connection between the word “Print” and the action can be changed by changing the environment
In response to Smith’s work
M.Wand and D.P.Frechman gave an explanation of Smith’s work using representational semantics.
After simplifying the programming language
After simplifying the programming language, the information about the level of operation is given in the form of meta-continuation data, which gives a display semantics.
It is possible to give some mathematical meaning to the data.
In the above approach, the meta-continuation goes out of the syntax, so the semantics goes out of the syntax. Translated with www.DeepL.com/Translator (free version)

A Sphere Theoretic Axiomatization of Algebraic Language Theory and Its Unification with Galois Theory / Takeo Uramoto

1Introduction
A branch of theoretical computer science called algebraic language theory has been
Through a sphere theory formulation
A new perspective on Galois theory in integer theory
Mathematically speaking
The core theorem of this phenomenon is the duality theorem between spheres and sub-semantic objects, called semigalois sphere.
Both algebraic language theory and Galois theory are unified by this duality theorem.
Sphere theory is
Reveals that seemingly different theories actually share a “common pattern
Explicitly talk about explicitly shared patterns
Formal study of two languages
What kind of theory is algebraic language theory?
A theory of the mathematical study of language
What is “language”?
A certain symbolic system defined to convey our thoughts, feelings, and will to others.
Research on language
There are various approaches such as cognitive science, psychology, philosophy, etc.
Formal language theory
Just as physics mathematically formalizes the world to describe the laws of the world
Algebraic theory is an approach to the study of language.
Among them, algebraic theory is
Algebraic theory is the study of language using algebraic theories from pure mathematics, such as finite semigroup theory.
Typical problems studied in formal language theory (and algebraic language theory) are
A typical problem studied in formal (and algebraic) language theory is formulated as a combinatorial problem concerning a finite set of strings.
Intuitive meaning is
In linguistics, it has its roots in the work of Noam Chomsky on generative grammar.
To put it simply
In general, a language is
formally defined as a set of finite strings, and
Formal language theory is based on the idea that
Formal language theory is primarily concerned with the symbolic patterns of language.
in the context of linguistics, which is primarily concerned with describing and classifying the structure of language.
In the context of linguistics
In the context of linguistics, it is related to the study of the “grammar” of language
It is also equivalent to computational logic problems.
English and Japanese are natural languages.
English and Japanese, which are natural languages, are not easy to find out their grammar.
It is not obvious to describe explicitly (i.e., to construct a grammar) what kind of sequence (or pattern) of characters is necessary for an English sentence to be valid.
To understand the meaning of a language
Focusing on certain patterns in a sentence
Chomsky’s generative grammar was mathematically defined and studied as a general notation for explicitly describing such patterns in language.
Generative grammars can describe the nested structure of sentences.
It has been applied to the design of programming languages.
A language described by Chomsky’s generative grammar is equivalent to a language that can be described by a computational model called a Turing machine.
In formal language theory
Formal language theory is a hierarchical, computational approach to language classification, focusing on the combinatorial patterns of sentences as simple symbolic strings, outside the context of linguistics.
In relation to the problem in linguistics, given a language L, what we are interested in is
Given a language L, what we are interested in is finding a pattern (grammar) of w that characterizes what kind of string w ∈ ∑* belongs to L.
If there is a generative grammar that characterizes the pattern of L
From the transparency between the generative grammar and the Turing machine
This means that there is an algorithm.
In algebraic language theory
Algebraic language theory applies the findings of algebra to the classification of language hierarchies.
Historically, it has been used since the 1960s.
The theory is limited to very simple languages, called “regular languages”.
Instead, it has a systematic methodology not found in the general language hierarchy.
Ideal foundation for formal language theory (or computation theory)
Linear algebra” in the theory of computation
3-Algebraic language theory–classification theory of regular languages
Define a regular language to be classified by the theory
A regular language is characterized as a language that can be described by a certain kind of expression called a “regular expression”.
In the case of Chomsky’s generative grammar
In Chomsky’s generative grammar, it is equivalent to a language describable by a type 3 generative grammar.
Generative grammars can be roughly divided into types 0 to 3, with type 3 being a market or simple hierarchy.
4The unification of Galois theory with sphere theory
5Conclusion

Mathematical Models of Software and Area Theory / Masayuki Hiyama

Introduction
Is sphere theory useful for software problems?
Yes, it is.
Provide a theoretical basis for software engineering/computational chemistry
To guide us in designing real programming languages and software systems
It provides a robust and consistent framework for understanding phenomena related to software.
What sphere-theoretic concepts are used and how are they used in relation to software?
The Cartesian sphere appears in the Cary-Howard-Lambeck correspondence, which maps lambda-calculus, logic, and the sphere.
The traced monoidal sphere is a sphere used in programming to formulate recursive methods or recursive phenomena.
Monads are tools for working with functions that do more than just pure computation.
Monads and their associated companions and can-extensions are also the backbone of functional design and programming methods in programming languages.
Spivak’s theory is a direct application (without customization) of sphere theory to databases.
The Yoneda embedding of sphere theory is the same as the continuation-me method of programming.
Set spheres, spheres of paths in directed graphs
A point represents some object, and an arrow represents some projection, when the object and projection of a sphere are Zushi.
A figure formed by a point and an arrow is called a directed graph.
Suppose we are given a directed graph.
A directed graph is not a sphere as it is.
It is easy to create a sphere from a directed graph.
A path from one point to another point
A path in a directed graph is written down by the following rules
1. write the starting point at the beginning and the arrival point at the end
2. write the arrows (labels) between the starting point and the arrival point, separated by commas.
3. enclose the whole thing in square brackets
Examples
1.[A,a,a,b,c,b,d,C] 2.
2.[B,c,a,b,B] 3.
3.[B,d,C] 4.
4.[B,c,a,a,b,c,b,B].
Paths that do not pass through the arrows at all are allowed
1.[A,A] 2.
2.[B,B] 3.
3.[C,C].
We can construct a sphere using points and paths of directed graphs (zero length is allowed).
Target is a point in a directed graph
Example
{A,B,C] The projections are the paths of the directed graph
Example
[B,c,a,b,B].
The set of projections is an infinite set
The domain of the projections [X,a,Y] is
X
The remainder of the projection [X,a,Y] is
Y
The composition of the two projections [X,α,Y] and [Y,β,Z] is
[X,α,β,Z].
The identity projection of the object X is
[X,X].
Since the composition operation of the path satisfies the computational laws of the sphere (union law, left unit law, right unit law)
This means that the sphere has been constructed.
We have a sphere composed of a directed graph.
We now have two spheres
1 Set sphere: a sphere where the object is a set and the projection is a map
2 The sphere of paths in directed graphs: where the object is a point and the projection is a path.
Abstract models of things and interfaces
As an example, we model a counting machine as a real world object
1. countup operation on a counter object raises the total by one
2. when you do a reset operation on a counter object, the total value goes to zero
3. value operation on the counter object returns the total value.
What is the difference between a real-world counter and a counter object that exists in the software world?
They do not come with user interface statements such as buttons or indicators.
The counter object abstracts only the internal functionality of the counting instrument.
The counter object holds the numerical value as an internal state.
To change the internal state externally, use the
countup operation
reset operation
The “value operation” is used to know the internal state of the object, but it does not change the internal state of the object.
Counter objects inside the computer are manipulated by countup, rest, and value.
The user, a person outside the computer, does not know how to manipulate the object.
Two boundaries come into play
“Description of the software model of a number cruncher” is the programming interface.
It does not include information about the user interface.
Interface Counter { countup() :void; reset() :void; value() :number; }
Signature Counter { sort S operation init:Void→S operation counter:S→S operation reset:S→S operation value:S→Nat }
When dealing with programming interfaces (specification descriptions) theoretically, they are called “signatures”.
Sort S
Let S be the set of internal states of the counter object.
Operation init:Void→S
The Init operation is a mapping from a unitary set (a set with a single element) Void to S.
This operation is used to initialize a counter object (a one-time setting at the beginning of use).
Operation counter:S→S
The Countup operation is a mapping from a set of internal states S to S.
This kind of mapping is called a “state transition”.
Operation reset:S→S
The Reset operation is a mapping from a set of internal states S to S.
Unlike the Init operation, it can be used at any time.
This mapping is also a state transition.
Operation value:S→Nat
The Value operation is a mapping from a set of internal states S to a set of natural numbers Nat.
Sekite as a conceptual implementation of the interface
Represented by a directed graph
Not an actual program (running on computer hardware)
Implement” as a conceptual entity
Seven symbols written as text: S, init, void, countup, reset, value, Nat
The seven symbols written as text: S, init, void, countup, reset, value, and Nat, are divided into two categories: fixed symbols (symbols whose meaning is fixed in advance) and variable symbols (symbols whose meaning is undefined).
Variant symbol S: denotes an undefined set 2. In mathematical writing, it promises 1. 1={0}. 3. constant symbol Nat: The meaning is defined and is the set of natural numbers. In mathematical writing, it is N. N={0,1,2,…} 4. Variant symbol init: Indicates an undefined mapping. 5. Variant symbol count: denotes an undefined map 6. Variant symbol reset: denotes an undefined map 7. Variant symbol value: denotes an undefined map
We will assign mathematical entities to these symbols.
Assign mathematical entities to these symbols. The expression “The meaning of ~ is M(~) = …” is rewritten as “M(~) = …”. where M is Meaning or Model.
S, Void, Nat are points of the directed graph, init, countup, reset, value are arrows of the directed graph.
M maps the points and arrows of the directed graph to sets and maps, respectively
The directed graph generates the sphere of the path
Correspondence M can be extended and defined for paths
A path can be interpreted as a kind of command sequence sent to a counter object
The actual action or behavior according to the command sequence is conceptually (mathematically) implied as a map corresponding to M
The correspondence M extended to a path is actually a Kante.
A Kante is a correspondence that
1. an object in a sphere is associated with an object in another or the same sphere
2. map a projectile of the sphere to a projectile of another or the same sphere
3. the composite (union) of two projectiles corresponds to the composite of the projectiles in the corresponding destination
4. the identity projection of an object is the identity projection of its corresponding destination object.
In fact, M has the following properties
1. maps a point of a directed graph, which is an object of the sphere of a path, to a set, which is an object of the sphere of a set
2. maps a path, which is a surjection of the sphere of a path, to a map, which is a surjection of the set sphere
A path that is a composition (union) of two paths is associated with a composition of mappings in the corresponding destination.
A path of length 0 from a point maps to the composition of the maps in the corresponding destination set at that point.
The rest goes like rabbits–indexed sphere/fibered sphere/institution
The Kante M defined in the previous section is
a natural and standard mathematical model of a number cruncher.
There is an upper limit to the number of digits in the form of a real number criterion.
Assume that the numerical value is a single digit (from 0 to 9)
1. m(1s)={0,1,2,…,9} 2. ,9} 2. M1(Void)=1={0} 3. M1(Nat)=N 4. M1(init)=map(1∋0 ↦ 0∈{0,1,2,…,9} 5. ,9} 5. M1(countup)=map({0,1,2,… ,9}∋ k ↦ (if(k=9) then 9 else k+1∈{0,1,2,…,9}) 6. ,9}) 6. M1(reset)=map({0,1,2,. ,9}∋ k ↦ 0∈{0,1,2,…,9}) 7. ,9}) 7. M1(value)=map({0,1,2,. ,9} ∋ k ↦ k∈N)
This conceptual counter stalls when the internal coefficient value reaches 9.
For a single indicator (an interface in programming terms, a directed graph in illustration)
There are many mathematical models of it.
No matter how you choose, the fact remains that the model is a function.
Let PathCat(Σ) be the sphere of the path created by considering the index Σ as a directed graph.
What is the model of an indicator Σ?
Even in the case of counters, there was not one but many “model = Kante”.
Let Functor(PathCat(Σ).Set) be the set of all possible “models = Kante”.
If we add a higher-order correspondence, called a natural transformation, to the set of factors
we can construct a new sphere.
Let FunctorCat(PathCat(Σ).Sey) be the arithmetical sphere.
This Kante sphere is the appropriate stage to deal with models of index Σ.
Also written as Model(Σ)
There is a huge number of indicators to deal with, and if we consider all possible indicators and the appropriate correspondence between them together, a sphere is again formed.
We call it a signature category.
We write Sig for the specifically constructed sphere of indicators.
Model (Σ) is a sphere, and Σ is an object of the sphere Sig.
If we move Σ, we get the correspondence Σ ↦ Model(Σ) maps the sphere to the object of the sphere Sig
A sphere is a “sphere of spheres” CAT object
Model(Σ) maps sphere CAT objects to sphere Sig objects
It is also possible to “map a projectile of the sphere CAT to a projectile of the sphere Sig, inverted.
Model becomes an antivariant function (a function that reverses the direction of the projection), Model:Sig →CAT.
An antivariant move that takes values in the “sphere of the sphere” is called an “indexed category.
By applying a procedure called Grothendieck construction
A fibered category is obtained by applying the Grothendieck construction procedure.
Appropriate additional functions can be added to the indexed/fibered category
By adding appropriate additional functions to the indexed/fibered sphere, it is possible to create a sphere theoretic structure called the institution of Joseph Goguen and Rod Burstall Translated with www.DeepL.com/Translator (free version)

Science/Art
Sphere Theory as a Written Language of Science / Shogo Tanimura

Notes
The Power of Numbers
Algebraization
Is a Mathematical Expression a Process or a Product?
Loose Congruence
What is possible with algebraization
Spatialization
Sphere
Three kinds of arrows
Theory of Quantity
Is Science Realism or Epistemology?

Universality and Its Fluctuation — Sphere Theory of Networks / Taichi Haruna

1Introduction
Network Theory as an Object
Which vertices and edges in a network are important?
More vertices and adjacent vertices are important
Vertices or edges through which more shortest paths pass are important
Vertices that are adjacent to important vertices are important
The internal structure of the network matters
Major concerns
Real-world network structure and dynamics in biological, social, and information networks
Quantifying the features of network structure
Mathematical models of networks that reproduce the quantified features
Sphere theory of networks
Consider networks as processes
Sphere theory of networks is proposed as a general framework for describing open networks.
Open networks are
have connections for input and output, such as electric circuits and chemical reaction networks
Open networks can be “synthesized” through their connections.
Connections are objects, open networks are projections
The main interest is
Not the internal structure of the network itself
but the behavior of the open network that can be observed from the outside
In the case of chemical reaction networks
The steady-state relationship between the concentration and influx of the input chemical species and the concentration and efflux of the output chemical species
Whether it is conserved in the synthesis or monodial product between open networks
Monodial product
The “juxtaposition” of connections or open networks
Composite
Serialize
The internal structure of the network is a black box
Seeing a network as a projectile
Networks as Things” and “Networks as Process
Networks as Process
2Intrinsic Development of the Sphere Theory of Networks
2-1Preparation
Discuss the Yoneda extension in terms of functions in directed networks.
Consider a field in a directed network as a process.
A directed network is a structure consisting of the following four sets
A set of vertices
A set of arrows
For a given arrow, there are two maps corresponding to its starting and ending vertices
2-2 Vertices are processes, and arrows are interfaces between vertices as processes
Consider a vertex as a process
Example
Gene regulatory network
Vertex is a gene
The vertex is regulated by other genes
Through transcription and translation of DNA, it synthesizes proteins that can regulate other genes
Arrows indicate regulatory relationships between genes
The arrows indicate the regulatory relationship between the genes, which can be viewed as a “process interface between the vertices.
Neural networks
Ecological network
Networks of organisms such as
2-3 Arrows are things, and vertices are constraints on arrows as things
Think of arrows as things
Example
Catalytic reaction network
What is a catalytic reaction?
Chemical reaction depicted by the chemical reaction equation C1+A→C1+C2
A is the substrate of the catalytic reaction
C2 is the product
C1 is the catalytic molecule
When C2 is also the catalytic molecule in a catalytic reaction
By drawing an arrow from C1 to C2
By drawing arrows from C1 to C2, we can see the relationship between the catalytic molecules of interest
and the relationship between them in the catalytic reaction is the arrow.
A directed network can be formed
In a catalytic reaction
In a catalytic reaction, the reaction proceeds only when the substrate of the reaction is incorporated.
The arrow of the catalytic network has the aspect of “substrate = thing”.
A vertex, which is a catalytic molecule
If there is an arrow going to it and an arrow going out of it
These arrows are the constraints that connect things.
The constraints here are
The constraint here acts on a change (in this case, the transformation of a substrate into a product) and
The aspect of the action that is of interest is the thing that is unaffected by the change.
Closure of constraints
A network of constraints, with arrows pointing to relationships in which one constraint generates another, is
A network of constraints is called a constraint closure if it satisfies the condition that there is an arrow going into any vertex.
It is called constraint closure.
2-4 Universality and Its Consequences I: Networks Opened from the Inside
Universality regarding the idea that “vertices are processes, and arrows are interfaces between vertices as processes.
For each vertex, by counting the number of lateral paths that pass through the vertex as the starting (or ending) point, we can obtain the mediation centrality that determines the importance of the vertex as an input (or output).
Lateral paths
A sequence of arrows that alternately share a viewpoint and an endpoint.
Rather than specifying the input and output parts of the network in advance, as in sphere theory network theory, we can use the internal structure of the network to determine the input and output parts.
Select input/output candidates from the internal structure of the network
Evaluate how the network can be opened from the inside.
2-5 Universality and Its Consequences II: Carriers of Network Self-Determinism
Universality of the idea that “arrows are things, and vertices are constraints on arrows as things
In a paper discussing whether Kauffman’s model of a catalytic reaction network consisting of random polymers (20) is a valid model for explaining the origin of life (21), it was argued that the strongly connected components of a catalytic reaction network can maintain themselves and their periphery as long as they are well supplied with substrate. In this paper (21), I hypothesize that the strongly linked component in the catalytic network corresponds approximately to the primitive genotype, and the periphery corresponds approximately to the primitive phenotype, because it can maintain itself and its periphery as long as the substrate is sufficiently supplied.
3 Fluctuations in Universality
3-1 Tightness Theorem and Varela’s Asymmetric Complementarity
In the chapter entitled “The Complementarity Framework” in his book “Principles of Biological Autonomy” (23), Varela states
In the chapter entitled “Complementarity Framework” in his book “Principles of Biological Autonomy” (23), Varela argues that it is useful for understanding biological and cognitive systems to treat the relationships between different descriptions of the system as complementary rather than contradictory
In doing so, he presents the notion of asymmetrical complementarity, including as an example, accompanying in sphere theory.
For example, the following pairs are mentioned: “whole/parts of a whole,” “existence/generation,” “environment/system,” and “autonomy/control.
These are collectively referred to as “theit/theprocessleadingtoit,” and it is emphasized that the two terms of the pair are not on the same level of equality, but are in an asymmetrical relationship of overlapping levels, where the first term encompasses the second. This is emphasized.
3-2 Adaptive network model evolving in time to a critical state
In general, biological systems face the problem of achieving both stable behavior in a given environment and adaptability to environmental changes.
In complex systems science, one solution is to bring the system state to a critical state at the boundary between stable and unstable states, and understanding the mechanism is a research issue.
4 Concluding Remarks — Random Kan Extensions
The operation of “replacing universals by individuals” applied to the tightness theorem in Section 3 can also be applied to more general Kan extensions. This is called the random Kan extension.
Based on the random Kan extension, we can construct models that realize three of the different mechanisms (critical state, preferential selective growth, and SSR) that produce the power law, respectively.
We can construct models that realize three different mechanisms (critical state, preferential selective growth, and SSR) for generating power law based on random Kan expansion. Translated with www.DeepL.com/Translator (free version)

The Development of the Theory of the Sphere – A Turn toward the Theory of the Desphere / Yukio Gunji Pegio

1Introduction
2Sphere theory: an approach that brings two opposites into a comparable pair
3The Deconstructed Sphere
3-1Deconstruction of the contrast condition
3-2Sphere Theory and Desphere Theory Approaches in Quantum Psychology
4 Conclusion

A Theory of Art from the Viewpoint of the Sphere Diagram — Holes, Cohomology, and Abduction / Akihiro Kubota

1What does it mean to be the same?
Inherence of topological space
2From the Euler characteristic to the homology group
3Abelian groups and quasi-homomorphic mappings
4Mathematical Models as Cognitive Forms
5Description of inference by homological algebra
6From sets to spheres
7The Arbelian sphere and the decomposition of projections
8The Art Nexus and the Sphere of Art
9Agency as a Kante
10What is a work of art?
11Figuralism and its intrinsic nature

Philosophy
Deepening Phenomenology through Sphere Theory — Monadology, Monadology, and Self / Shigeru Taguchi + Kouyato Saigo

Introduction
1 Monism of projectiles: Sphere theory and the mediationist view of reality
2 Mediational and Spherological Interpretations of “Being the Same
3A Spherological Representation of the Relationship between Self and Others: Using the Slice Sphere as a Guide
Introduction
The Slice Sphere
Self-other relations
4From Monadology to “Self as Mediator
Leibniz’s Monadology
Monad and the World in Husserl
Can We Know the Original Sphere from the Slice Sphere?
The Self Seeping to the Bottom of the First Person
Acknowledgments

What understanding does the structural concept of mathematics bring to French structuralism — Bourbaki, Cavaillès, Rotman, and the theory of the sphere as a guide / Daisuke Nakamura

Introduction
1 Reconsidering Bourbaki’s concept of “structure
2The Concept of “Structure” and the “Renewal” of Academic Knowledge in Kavayes
3Lotman: Bourbaki’s “Esotericism” (?) and its Resonance with Sphere Theory The Resonance of Bourbaki’s “Esotericism” and Sphere Theory
Conclusion

A Critical Study of the Relationship between Alain Badiou’s Philosophy and Mathematics: Aspects of the Post-Cavaillès Development of the “Philosophy of Concepts” / Kazutaka Kondo

Badiou and Sphere Theory
Cavayes’s “Philosophy of Concepts” and its later development
On Badiou’s Reference to the Theory of the Sphere, Especially to Topos Theory
Conclusion and Discussion

Serials ● A Scientist’s Walk ● Vol. 69
In Search of a New Home — Character Education and Science / Fumitaka Sato

Viruses are a strategy for integrating the humanities and sciences
The impetus for the separation of arts and sciences
Is science a soldier?
A large young man of all ages
Gender differences, masters, Asianization…
A Look at the History of German National Education
The Two Benefits of Science: Application and Character Education
Nation Building through Education
Pure mathematics is responsible for schooling
Professionalism
The Habermas Ruling
The disappearance of correctness
Science and character education
Nature and Man
Poetry is born when you look at Hiroshima.
Why do we think of the Big Bang?

The Research Notebook
Human Differences and Today’s Phenomenology / Maiko Sakai