Tips for Beginners: Exploiting the quadratic function

1962 ◽  
Vol 55 (4) ◽  
pp. 299-301
Author(s):  
Lowell Leake
Keyword(s):  
Set Theory ◽  
Subject Matter ◽  
Mathematical Theory ◽  
Quadratic Function ◽  
Analytic Geometry ◽  
Mathematics Class

The quadratic function offers some unusual opportunities to assemble several segments of mathematical theory chosen from algebra, analytic geometry, and set theory. The exploitation of these opportunities invariably arouses the interest of students in any mathematics class having the proper background, and gives them an elegant vignette of the unity of subject matter in mathematics.

scholarly journals The World Is Not a Theorem

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1467
Author(s):  
Stuart Kauffman ◽  
Andrea Roli
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Generative Process ◽  
Behavioral Adaptations ◽  
The World

The evolution of the biosphere unfolds as a luxuriant generative process of new living forms and functions. Organisms adapt to their environment, exploit novel opportunities that are created in this continuous blooming dynamics. Affordances play a fundamental role in the evolution of the biosphere, for organisms can exploit them for new morphological and behavioral adaptations achieved by heritable variations and selection. This way, the opportunities offered by affordances are then actualized as ever novel adaptations. In this paper, we maintain that affordances elude a formalization that relies on set theory: we argue that it is not possible to apply set theory to affordances; therefore, we cannot devise a set-based mathematical theory to deduce the diachronic evolution of the biosphere.

scholarly journals Applications of Fuzzy Technology in Business Intelligence

2011 ◽  
Vol 6 (3) ◽  
pp. 428 ◽  
Author(s):  
Andreas Meyer ◽  
Hans-Jürgen Zimmermann
Keyword(s):  
Set Theory ◽  
Fuzzy Clustering ◽  
Fuzzy Set ◽  
Business Intelligence ◽  
Fuzzy Set Theory ◽  
Mathematical Theory ◽  
Business Management ◽  
Knowledge Based Systems ◽  
Knowledge Based ◽  
Single Method

Fuzzy Set Theory has been developed during the last decades to a demanding mathematical theory. There exist more than 50,000 publications in this area by now. Unluckily the number of reports on applications of fuzzy technology has become very scarce. The reasons for that are manifold: Real applications are normally not single-method-applications but rather complex combinations of different techniques, which are not suited for a publication in a journal. Sometimes considerations of competition my play a role, and sometimes the theoretical core of an application is not suited for publication. In this paper we shall focus on applications of fuzzy technology on real problems in business management. Two versions of fuzzy technology will be used: Fuzzy Knowledge based systems and fuzzy clustering. It is assumed that the reader is familiar with basic fuzzy set theory and the goal of the paper is, to show that the potential of applying fuzzy technology in management is still very large and hardly exploited so far.

Type-theoretical checking and philosophy of Mathematics

1998 ◽  
Author(s):  
Nicolaas Govert de Bruijn
Keyword(s):  
Set Theory ◽  
Subject Matter ◽  
Mutual Influence ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
General Information ◽  
Point Of View ◽  
De Bruijn ◽  
Level Of Understanding ◽  
The Way

After millennia of mathematics we have reached a level of understanding that can be represented physically. Humankind has managed to disentangle the intricate mixture of language, metalanguage and interpretation, isolating a body of formal, abstract mathematics that can be completely verified by machines. Systems for computer-aided verification have philosophical aspects. The design and usage of such systems are influenced by the way we think about mathematics, but it also works the other way. A number of aspects of this mutual influence will be discussed in this paper. In particular, attention will be given to philosophical aspects of type-theoretical systems. These definitely call for new attitudes: throughout the twentieth century most mathematicians had been trained to think in terms of untyped sets. The word “philosophy” will be used lightheartedly. It does not refer to serious professional philosophy, but just to meditation about the way one does one’s job. What used to be called philosophy of mathematics in the past was for a large part subject oriented. Most people characterized mathematics by its subject matter, classifying it as the science of space and number. From the verification system’s point of view, however, subject matter is irrelevant. Verification is involved with the rules of mathematical reasoning, not with the subject. The picture may be a bit confused, however, by the fact that so many people consider set theory, in particular untyped set theory, as part of the language and foundation of mathematics, rather than as a particular subject treated by mathematics. The views expressed in this paper are quite personal, and can mainly be carried back to the author’s design of the Automath system in the late 1960s, where the way to look upon the meaning (philosophy) of mathematics is inspired by the usage of the unification system and vice versa. See de Bruijn 1994b for various philosophical items concerning Automath, and Nederpelt et al. 1994, de Bruin 1980, de Bruijn 1991a for general information about the Automath project. Some of the points of view given in this paper are matters of taste, but most of them were imposed by the task of letting a machine follow what we say, a machine without any knowledge of our mathematical culture and without any knowledge of physical laws.

Probability, interpretations of

2018 ◽  
Author(s):  
Paul Humphreys
Keyword(s):  
Subject Matter ◽  
Relative Frequency ◽  
Mathematical Theory ◽  
Logical Probability ◽  
Philosophical Discourse ◽  
Radioactive Atom ◽  
Football Team ◽  
Inductive Support

The term ‘probability’ and its cognates occur frequently in both everyday and philosophical discourse. Unlike many other concepts, it is unprofitable to view ‘probability’ as having a unique meaning. Instead, there exist a number of distinct, albeit related, concepts, of which we here mention five: the classical or equiprobable view, the relative frequency view, the subjectivist or personalist view, the propensity view, and the logical probability view. None of these captures all of our legitimate uses of the term ‘probability’, which range from the clearly subjective, as in our assessment of the likelihood of one football team beating another, through the inferential, as when one set of sentences lends a degree of inductive support to another sentence, to the obviously objective, as in the physical chance of a radioactive atom decaying in the next minute. It is often said that what all these interpretations have in common is that they are all described by the same simple mathematical theory – ‘the theory of probability’ to be found in most elementary probability textbooks – and it has traditionally been the task of any interpretation to conform to that theory. But this saying does not hold up under closer examination, and it is better to consider each approach as dealing with a separate subject matter, the structure of which determines the structure of the appropriate calculus.

An introduction to γ-recursion theory (or what to do in KP – Foundation)

1990 ◽  
Vol 55 (1) ◽  
pp. 194-206 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Recursion Theory ◽  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Turing Degrees ◽  
The Subject ◽  
Admissible Ordinals ◽  
Limited Induction ◽  
Well Foundedness

The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)As go the integers so go the ordinals. One motivation of α-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of ω, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of α-recursion theory.When working in the theory KP – Foundation Schema (hereinafter called KP−), one should really not call it α-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name β-recursion theory refers to what would be α-recursion theory only it includes also inadmissible ordinals, we call the subject of study here γ-recursion theory. This answers a question by Sacks and S. Friedman, “What is γ-recursion theory?”

scholarly journals L’infini entre deux bouts. Dualités, univers algébriques, esquisses, diagrammes

2020 ◽  
Vol 41 (2) ◽  
Author(s):  
René Guitart
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Mathematical Thinking ◽  
Mathematical Work ◽  
Mathematical Activity ◽  
Actual Infinity ◽  
Zeno's Paradoxes ◽  
Mathematical Problems ◽  
Technical Details ◽  
The One

The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infinite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical problems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of mathematical work. From this perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality ― “between two ends”, as our title states ― that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infinity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises.

scholarly journals Исчисление касательных и вокруг

2018 ◽  
Author(s):  
A.G. Kusraev ◽  
S.S. Kutateladze
Keyword(s):  
Set Theory ◽  
Variational Analysis ◽  
Mathematical Theory ◽  
Local Analysis ◽  
Modern Approach ◽  
Standard Models ◽  
Models Of Set Theory

Optimization is the choice of what is most preferable. Geometry and local analysis of nonsmooth objects are needed for variational analysis which embraces optimization. These involved admissible directions and tangents as the limiting positions of the former. The calculus of tangents is one of the main techniques of optimization. Calculus reduces forecast to numbers, which is scalarization in modern parlance. Spontaneous solutions are often labile and rarely optimal. Thus, optimization as well as calculus of tangents deals with inequality, scalarization and stability. The purpose of this article is to give an overview of the modern approach to this range of questions based on non-standard models of set theory. A model of a mathematical theory is usually called nonstandard if the membership within the model has interpretation different from that of set theory. In the recent decades much research is done into the nonstandard methods located at the junctions of analysis and logic. This area requires the study of some new opportunities of modeling that open broad vistas for consideration and solution of various theoretical and applied problems.

scholarly journals On a Theory Objects Based on a Single Axiom Scheme

1966 ◽  
Vol 26 ◽  
pp. 13-30
Author(s):  
Katuzi Ono
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Axiom Of Choice ◽  
Fundamental Theory ◽  
Axiom Scheme ◽  
Single Axiom ◽  
The Axiom Of Choice ◽  
Formal Theories

There are some fundamental mathematical theories, such as the Fraenkel set-theory and the Bernays-Gödel set-theory, in which, I believe, all the actually important formal theories of mathematics can be embedded. Formal theories come into existence by being shown their consistency. As far as this is admitted, not all the axioms of set theory are necessary for a fundamental mathematical theory. The fundierung axiom is proved consistent by v. Neumann, the axiom of extensionality is proved consistent by Gandy, and even the axiom of choice is proved consistent by Göldel. Although it is not evident that a set-theory does not cease from being a fundamental theory of mathematics after abandoning these axioms all at once, the theory must be enough for being a fundamental theory of mathematics without some of them.

scholarly journals HIMPUNAN LEMBUT KABUR HESITANT DAN APLIKASINYA DALAM PENGAMBILAN KEPUTUSAN

2017 ◽  
Vol 6 (3) ◽  
pp. 23
Author(s):  
Sri Delvia Oriza ◽  
Nova Noliza Bakar
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Mathematical Theory ◽  
Soft Set ◽  
Hesitant Fuzzy Set ◽  
Mathematical Tool ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
Fuzzy Soft Sets ◽  
Level Soft Set

Abstract. Molodstov's soft set theory is a newly emerging mathematical tool to handleuncertainty. The soft set theory can be combined with other mathematical theory like asfuzzy set theory. This paper aims to extend hesitant fuzzy set to hesitant fuzzy soft sets.Then, the complement, "AND", "OR", union, intersection operations and De Morgan'slaw are dened on hesitant fuzzy soft sets. Finally, with the help of level soft set, thehesitant fuzzy soft sets are applied to a decision making problem.Kata Kunci: Soft set, Fuzzy set, Hesitant fuzzy set, Hesitant fuzzy soft set, Level softset

Mathematical Thinking: The Struggle for Meaning

1984 ◽  
Vol 15 (1) ◽  
pp. 35-49 ◽  
Author(s):  
Leone Burton
Keyword(s):  
Subject Matter ◽  
Mathematical Thinking ◽  
Descriptive Model ◽  
The Subject ◽  
Mathematics Class ◽  
Classroom Presentation

This paper argues that mathematical thinking is not thinking about the subject matter of mathematics but a style of thinking that is a function of panicular operations, processes, and dynamics recognizably mathematical. It further suggests that because mathematical thinking becomes confused with thinking about mathematics, there has been little success in separating process from content in the classroom presentation of the subject. A descriptive model of mathematical thinking is presented and then used to provide a practical response to the questions, Can mathematical thinking be taught? In what ways? The reacher is encouraged to recognize both what constitutes mathematical thinking, whether in the mathematics class or some other, and what conditions are necessary to foster it.

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