Constructive assertions in an extension of classical mathematics

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order on all finite groups, that is well behaved with respect to cardinality. In fact, if $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G$. Internally, there is also a canonical order for the elements of any finite group $G$, and we find equivalent objects. This allows us to find the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and a minimal set of independent equations that define the group is obtained. Examples are given, using all groups with less than ten elements, to illustrate the procedure for finding all groups of $n$ elements, and we order them externally and internally. The canonical block form of the symmetry group $\Delta_4$ is given and we find its automorphisms. These results are extended to the infinite case. A real number is an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, also. We make brief mention on the calculus of real numbers. In general, we are able to represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects of all types are well assigned to tree structures. We conclude with comments on type theory and future work on computational and physical aspects of these representations.

Download Full-text

The Problem of Mathematical Objects

Essence and Existence ◽

10.1093/oso/9780198854296.003.0013 ◽

2020 ◽

pp. 213-224

Author(s):

Bob Hale

Keyword(s):

Natural Numbers ◽

Real Numbers ◽

Mathematical Objects ◽

Epistemological Foundation ◽

Fundamental Part ◽

Object Based

If fundamental mathematical theories such as arithmetic and analysis are taken at face value, any attempt to provide an epistemological foundation—roughly, an account which explains how we can know standard mathematical theories to be true, or at least justifiably believe them—must confront the problem of mathematical objects—the problem of explaining how a belief in the existence of an infinity of natural numbers, an uncountable infinity of real numbers, etc., is to be justified. One small but fundamental part of the problem is discussed: whether we can be justified in believing that there is a denumerable infinity of natural numbers, or, more generally, an infinity of objects of any kind. The chapter considers two broad approaches to this problem—what are called object-based and property-based approaches.

Download Full-text

Canonical Set Theory for Classic Mathematics: A Linear Order on Finite Groups and Structures, with a Natural Extension to Infinite Structures

10.20944/preprints202007.0415.v1 ◽

2020 ◽

Author(s):

Juan Ramirez

Keyword(s):

Finite Group ◽

Finite Groups ◽

Linear Order ◽

Natural Extension ◽

Natural Numbers ◽

Real Numbers ◽

Tree Structures ◽

Mathematical Objects ◽

Finite Group Theory ◽

Block Form

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order on all finite groups, that is well behaved with respect to cardinality. In fact, if $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G$. There is also a canonical order for the elements of $G$ and we can define equivalent objects of $G$. This allows us to find the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and a minimal set of independent equations that define the group is obtained. We show how to find all groups of order $n$, and order them. Examples are given using all groups with order smaller than $10$. The canonical block form of the symmetry group $\Delta_4$ is given and we find its automorphisms. These results are extended to the infinite case. A real number is an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, as well. We make brief comments on treating the calculus of real numbers. In general, we represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects are well assigned to tree structures. We conclude with brief comments on type theory and future work on computational and physical aspects of these representations.

Download Full-text

Canonical Description of Group Theory: A Linear Order on All Finite Groups

10.20944/preprints202006.0098.v2 ◽

2020 ◽

Author(s):

Juan Pablo Ramirez

Keyword(s):

Finite Group ◽

Group Theory ◽

Natural Number ◽

Finite Groups ◽

Linear Order ◽

Natural Numbers ◽

Real Numbers ◽

Good Definition ◽

Mathematical Objects ◽

Real Functions

We provide a construction of natural numbers that is unique with respect to other constructions, and use this construction in the domain of algebra and finite functions to find several results in finite group theory. First, we give a linear order to the set of all finite functions. This gives a linear order in the subset of all finite permutations. To do this, we assign a unique natural number, $N_f$, to every finite function $f$. The sub order on permutations is well defined with respect to cardinality; if $\eta_m,\eta_n$ are permutations on $m<n$ objects, then $N_{\eta_m}<N_{\eta_n}$. This representation also has the characteristic $N_{\textbf{1}_n}<N_{\eta}<N_{\textbf{id}_n}$ where $\textbf{1}_n$ is the one-cycle permutation of $n$ objects, $\textbf{id}_n$ is the identity permutation of $n$ objects, and $\eta$ is any permutation of $n$ objects. This representation provides a good definition of equivalent functions, and equivalent objects on functions. We are able to do this for both concrete functions, and abstract functions. We use this in the main section, on group theory, to number the set of all finite groups. We are able to well represent every finite group as a natural number; two groups are represented by the same natural number if and only if they are in the same isomorphism class. In fact, we are able to give a linear order to the set of finite groups. Specifically, we give a canonical bijective function $\textbf{G}_{Fin}\rightarrow\mathbb N$. This representation, $N_G$, of $G$, is also well behaved with respect to cardinality. Additionally, the cyclic group $\mathbb Z_n$ has smaller representation than any group of $n$ objects, and the group with largest representation is the abelian group $\mathbb Z_{p_1}^{n_1}\oplus\mathbb Z_{p_2}^{n_2}\oplus\cdots\oplus\mathbb Z_{p_k}^{n_k}$, where $n=p_1^{n_1}p_2^{n_2}\cdots p_{k}^{n_k}$ is the prime factorization of $n$. This representation of a finite group as a natural number also provides a linear order to the elements of the group, arranging its Cayley table in a canonical block form. The last section is an introductory description of real numbers as infinite sets of natural numbers. Real functions are represented as sets of real numbers, and sequences of real functions $f_1,f_2,\ldots$ are well represented by sets of real numbers, as well. In the last section we well assign mathematical objects to tree structures and conclude with some brief comments on type theory and future work. In general we are able to represent and manipulate mathematical objects with the smallest possible type, and minimum complexity.

Download Full-text

Canonical Description of Group Theory: A Linear Order on All Finite Groups

10.20944/preprints202006.0098.v3 ◽

2020 ◽

Author(s):

Juan Pablo Ramirez

Keyword(s):

Finite Group ◽

Group Theory ◽

Finite Groups ◽

Linear Order ◽

Natural Numbers ◽

Real Numbers ◽

Prime Factorization ◽

Tree Structures ◽

Mathematical Objects ◽

Block Form

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order, on the quotient space of isomorphism classes of finite groups, that is well behaved with respect to cardinality. If $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G\leq\mathbb Z_{p_1}^{n_1}\oplus\mathbb Z_{p_2}^{n_2}\oplus\cdots\oplus\mathbb Z_{p_k}^{n_k}$ where $n=p_1^{n_1}p_2^{n_2}\cdots p_{k}^{n_k}$ is the prime factorization of $n$. We find a canonical order for the objects of $G$ and define equivalent objects of $G$, thus finding the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and we are provided with a minimal set of independent equations that define the group. We show how to find all groups of order $n$, and order them. We give examples using all groups with order smaller than $10$, and we find the canonical block form of the symmetry group $\Delta_4$. In the next section, we extend our results to the infinite case, which defines a real number as an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, as well. In general, we represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects are well assigned to tree structures. We conclude with brief comments on type theory and future work on computational aspects of these representations.

Download Full-text

A New Set Theory for Analysis

Axioms ◽

10.3390/axioms8010031 ◽

2019 ◽

Vol 8 (1) ◽

pp. 31

Author(s):

Juan Ramírez

Keyword(s):

Real Number ◽

Number System ◽

Order Structure ◽

Natural Numbers ◽

Real Numbers ◽

The Real ◽

Power Set ◽

Universe Of Sets ◽

The Universe ◽

Real Number System

We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

Download Full-text

Canonical Set Theory for Classic Mathematics: A Linear Order on Finite Groups and Structures, with a Natural Extension to Infinite Structures

10.20944/preprints202007.0415.v2 ◽

2020 ◽

Author(s):

Juan Ramirez

Keyword(s):

Finite Group ◽

Finite Groups ◽

Linear Order ◽

Natural Extension ◽

Natural Numbers ◽

Real Numbers ◽

Tree Structures ◽

Mathematical Objects ◽

Finite Group Theory ◽

Block Form

We provide an axiomatic base for the set of natural numbers, that has been proposed as a canonical construction, and use this definition of $\mathbb N$ to find several results on finite group theory. Every finite group $G$, is well represented with a natural number $N_G$; if $N_G=N_H$ then $H,G$ are in the same isomorphism class. We have a linear order on all finite groups, that is well behaved with respect to cardinality. In fact, if $H,G$ are two finite groups such that $|H|=m<n=|G|$, then $H<\mathbb Z_n\leq G$. There is also a canonical order for the elements of $G$ and we can define equivalent objects of $G$. This allows us to find the automorphisms of $G$. The Cayley table of $G$ takes canonical block form, and a minimal set of independent equations that define the group is obtained. We show how to find all groups of order $n$, and order them. Examples are given using all groups with order smaller than $10$. The canonical block form of the symmetry group $\Delta_4$ is given and we find its automorphisms. These results are extended to the infinite case. A real number is an infinite set of natural numbers. A real function is a set of real numbers, and a sequence of real functions $f_1,f_2,\ldots$ is well represented by a set of real numbers, as well. We make brief comments on treating the calculus of real numbers. In general, we represent mathematical objects using the smallest possible data-type. In the last section, mathematical objects are well assigned to tree structures. We conclude with brief comments on type theory and future work on computational and physical aspects of these representations.

Download Full-text

Canonical Description of Group Theory: A Linear Order on All Finite Groups

10.20944/preprints202006.0098.v1 ◽

2020 ◽

Author(s):

Juan Pablo Ramirez

Keyword(s):

Finite Group ◽

Group Theory ◽

Natural Number ◽

Finite Groups ◽

Linear Order ◽

Natural Numbers ◽

Real Numbers ◽

Good Definition ◽

Mathematical Objects ◽

Real Functions

We provide a construction of natural numbers that is unique with respect to other constructions, and use this construction in the domain of algebra and finite functions to find several results in finite group theory. First, we give a linear order to the set of all finite functions. This gives a linear order in the subset of all finite permutations. To do this, we assign a unique natural number, $N_f$, to every finite function $f$. The sub order on permutations is well defined with respect to cardinality; if $\eta_m,\eta_n$ are permutations on $m<n$ objects, then $N_{\eta_m}<N_{\eta_n}$. This representation also has the characteristic $N_{\textbf{1}_n}<N_{\eta}<N_{\textbf{id}_n}$ where $\textbf{1}_n$ is the one-cycle permutation of $n$ objects, $\textbf{id}_n$ is the identity permutation of $n$ objects, and $\eta$ is any permutation of $n$ objects. This representation provides a good definition of equivalent functions, and equivalent objects on functions. We are able to do this for both concrete functions, and abstract functions. We use this in the main section, on group theory, to number the set of all finite groups. We are able to well represent every finite group as a natural number; two groups are represented by the same natural number if and only if they are in the same isomorphism class. In fact, we are able to give a linear order to the set of finite groups. Specifically, we give a canonical bijective function $\textbf{G}_{Fin}\rightarrow\mathbb N$. This representation, $N_G$, of $G$, is also well behaved with respect to cardinality. Additionally, the cyclic group $\mathbb Z_n$ has smaller representation than any group of $n$ objects, and the group with largest representation is the abelian group $\mathbb Z_{p_1}^{n_1}\oplus\mathbb Z_{p_2}^{n_2}\oplus\cdots\oplus\mathbb Z_{p_k}^{n_k}$, where $n=p_1^{n_1}p_2^{n_2}\cdots p_{k}^{n_k}$ is the prime factorization of $n$. This representation of a finite group as a natural number also provides a linear order to the elements of the group, arranging its Cayley table in a canonical block form. The last section is an introductory description of real numbers as infinite sets of natural numbers. Real functions are represented as sets of real numbers, and sequences of real functions $f_1,f_2,\ldots$ are well represented by sets of real numbers, as well. In the last section we well assign mathematical objects to tree structures and conclude with some brief comments on type theory and future work. In general we are able to represent and manipulate mathematical objects with the smallest possible type, and minimum complexity.

Download Full-text

Juliana Apostaty mit o Heliosie

Vox Patrum ◽

10.31743/vp.4351 ◽

2010 ◽

Vol 55 ◽

pp. 477-498

Author(s):

Ewa Osek

Keyword(s):

Detailed Analysis ◽

The One ◽

The Mind ◽

The Universe

The present paper is a brief study on Julian the Apostate’s religion with the detailed analysis of the so called Helios myth being a part of his speech Against Heraclius (Or. VII), delivered in Constantinople in AD 362. In the chapter one I discuss veracity of the Gregory of Nazianzus’ account in the Contra Julianum (Or. IV-V) on the emperor’s strange Gods and cults. In the chapter two the reconstruction of the Julian’s theological system has been presented and the place of Helios in this hierarchy has been shown. The chapter three consists of the short preface to the Against Heraclius and of the appendix with the Polish translation and commentary on the Julian’s Helios myth. The Emperor’s theosophy, known from his four orations (X-XI and VII-VIII), bears an imprint of the Jamblichean speculation on it. The gods are arranged in the three neo-Platonic hypostases: the One, the Mind, and the Soul, named Zeus, Hecate, and Sarapis. The second and third hypostases contain in themselves the enneads and the triads. The Helios’ position is between the noetic world and the cosmic gods, so he becomes a mediator or a centre of the universe and he is assimilated with Zeus the Highest God as well as with the subordinated gods like Apollo, Dionysus, Sarapis, and Hermes. The King Helios was also the Emperor’s personal God, who saved him from the danger of death in AD 337 and 350. These tragic events are described by Julian in the allegorical fable (Or. VII 22). The question is who was Helios of the Julian’s myth: the noetic God, the Hellenistic Helios, the Persian Mithras, the Chaldean fire, or the Orphic Phanes, what is suggested by the Gregory’s invective. The answer is that the King Helios was all of them. The Helios myth in Or. VII is the best illustration of the extreme syncretism of the Julian’s heliolatry, where the neo-Platonic, Hellenistic, magic, and Persian components are mingled.

Download Full-text

“Stonehenge in the Mind” and “Stonehenge on the Ground”:

Journal of the Society of Architectural Historians ◽

10.1525/jsah.2018.77.3.285 ◽

2018 ◽

Vol 77 (3) ◽

pp. 285-299 ◽

Cited By ~ 1

Author(s):

Ryan Roark

Keyword(s):

Subject Matter ◽

Formal Analysis ◽

Original Text ◽

Existing Structures ◽

The Subject ◽

The Mind

In Stone-Heng Restored (1655), Inigo Jones, the father of English neoclassicism, used drawings, histories, and questionable logic to argue that Stonehenge was built by the ancient Romans and that it originally exhibited perfect Platonic geometries. This argument was never given much credence, but by 1725 the subject matter and the architect had received enough attention that two book-length responses (a challenge and a defense) were published, and both were then republished in a single volume alongside Jones's original text. While most Jones scholars have neglected this work because of its logical and historical shortcomings, Ryan Roark argues in “Stonehenge in the Mind” and “Stonehenge on the Ground”: Reader, Viewer, and Object in Inigo Jones's Stone-Heng Restored (1655) that it was in fact exemplary of what made Jones, for many, a protomodern architect and scholar. Rather than viewing Jones's book as an earnest attempt to prove a historical inaccuracy, Roark considers it as an exercise in formal analysis, one that set the precedent for the contemporary pedagogical trend of using geometric simplifications of existing structures as a first step in new design. Jones's idiosyncratic reading of Stonehenge belied the idea that such analysis could be anything but intensely reliant on the subjectivity of both architect and viewer.

Download Full-text