Arithmetic, philosophical issues in

2018 ◽  
Author(s):  
Michael Potter
Keyword(s):  
Natural Number ◽  
Decision Procedure ◽  
Mathematical Induction ◽  
Universal Quantifier ◽  
Number Concept ◽  
Special Character ◽  
The Status ◽  
Proof By Induction ◽  
Philosophy Of Arithmetic ◽  
Principle Of Mathematical Induction

The philosophy of arithmetic gains its special character from issues arising out of the status of the principle of mathematical induction. Indeed, it is just at the point where proof by induction enters that arithmetic stops being trivial. The propositions of elementary arithmetic – quantifier-free sentences such as ‘7+5=12’ – can be decided mechanically: once we know the rules for calculating, it is hard to see what mathematical interest can remain. As soon as we allow sentences with one universal quantifier, however – sentences of the form ‘(∀x)f(x)=0’ – we have no decision procedure either in principle or in practice, and can state some of the most profound and difficult problems in mathematics. (Goldbach’s conjecture that every even number greater than 2 is the sum of two primes, formulated in 1742 and still unsolved, is of this type.) It seems natural to regard as part of what we mean by natural numbers that they should obey the principle of induction. But this exhibits a form of circularity known as ‘impredicativity’: the statement of the principle involves quantification over properties of numbers, but to understand this quantification we must assume a prior grasp of the number concept, which it was our intention to define. It is nowadays a commonplace to draw a distinction between impredicative definitions, which are illegitimate, and impredicative specifications, which are not. The conclusion we should draw in this case is that the principle of induction on its own does not provide a non-circular route to an understanding of the natural number concept. We therefore need an independent argument. Four broad strategies have been attempted, which we shall consider in turn.

A Lexicographic Decision Rule With Tolerances

2001 ◽  
Vol 23 (2) ◽  
Author(s):  
Marlies Ahlert ◽  
Hartmut Kliemt
Keyword(s):  
Decision Rule ◽  
Decision Procedure ◽  
Status Quo ◽  
Organ Allocation ◽  
Choice Making ◽  
The Status

AbstractThe implementation of the Wujciak algorithm as a new rule for organ allocation by Eurotransplant is of considerable interest for the theorist of choice making. In the process reformers accepted the status quo in principle but expected that their potential opponents would be willing to make minimal or 'tolerable' concessions. Thereby the consensual introduction of new dimensions of value and reforms of allocation practices based thereupon became viable. The paper characterizes a decision procedure based on ‘almost lexicographically pre-ordering established values and practices’ in a stylized manner, presents a formal reconstruction of it and points out some of its potential implications for rule choices in general.

scholarly journals Bicameral Conflict Resolution in the European Union: An Empirical Analysis of Conciliation Committee Bargains

2007 ◽  
Vol 37 (2) ◽  
pp. 281-312 ◽  
Author(s):  
THOMAS KÖNIG ◽  
BJORN LINDBERG ◽  
SANDRA LECHNER ◽  
WINFRIED POHLMEIER
Keyword(s):  
European Union ◽  
Conflict Resolution ◽  
Power Distribution ◽  
Decision Procedure ◽  
European Parliament ◽  
Status Quo ◽  
The European Union ◽  
The Status ◽  
Institutional Actors ◽  
The Impact

This article is a study of bicameral conflict resolution between the Council and the European Parliament in the European Union, which has established a bicameral conciliation process under the co-decision procedure. Scholars commonly agree that the European Parliament has gained power under the co-decision procedure, but the impact of the conciliation process on the power distribution between the Council and the European Parliament remains unclear. The scholarly debate suggests that the power of the institutional actors depends on their proximity to the status quo, the (im-)patience and the specific preference distribution of the institutional actors, although most analyses assume that the Commission plays an insignificant role. Using an ordered probit model, this study examines the power distribution between the two institutional actors, the factors for their bargaining success and the role of the Commission in the period between 1999 and 2002. The findings show that the European Parliament wins most conflicts, but that the Council is more successful in multi-dimensional disputes. The results confirm some theoretical claims made in the literature, such as the importance of the status quo location and of preference cohesiveness. However, they also reject a major assumption in the literature on the irrelevance of the Commission in the conciliation process, which we show to have an influential informational position for parliamentary success.

A cut-free Gentzen-type system for the modal logic S5

1980 ◽  
Vol 45 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Masahiko Sato
Keyword(s):  
Modal Logic ◽  
Natural Number ◽  
Decision Procedure ◽  
Type System ◽  
Point Of View ◽  
Usual Sense ◽  
Cut Elimination ◽  
Strict Sense ◽  
Theoretic Point ◽  
Subformula Property

The modal logic S5 has been formulated in Gentzen-style by several authors such as Ohnishi and Matsumoto [4], Kanger [2], Mints [3] and Sato [5]. The system by Ohnishi and Matsumoto is natural, but the cut-elimination theorem in it fails to hold. Kanger's system enjoys cut-elimination theorem, but, strictly speaking, it is not a Gentzen-type system since each formula in a sequent is indexed by a natural number. The system S5+ of Mints is also cut-free, and its cut-elimination theorem is proved constructively via the cut-elimination theorem of Gentzen's LK. However, one of his rules does not have the so-called subformula property, which is desirable from the proof-theoretic point of view. Our system in [5] also enjoys the cut-elimination theorem. However, it is also not a Gentzen-type system in the strict sense, since each sequent in this system consists of a pair of sequents in the usual sense.In the present paper, we give a Gentzen-type system for S5 and prove the cut-elimination theorem in a constructive way. A decision procedure for S5 can be obtained as a by-product.The author wishes to thank the referee for pointing out some errors in the first version of the paper as well as for his suggestions which improved the readability of the paper.

Reviews and Evaluations

1968 ◽  
Vol 61 (1) ◽  
pp. 86-88
Author(s):  
Harold Tinnappel
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
School Teacher ◽  
High School Teacher ◽  
Mathematical Induction ◽  
Order Property ◽  
Positive Real ◽  
Induction Argument ◽  
Twelfth Grade ◽  
Proof By Induction

Although this is a calculus text, the first two chapters might be of interest to teachers of eleventh- and twelfth-grade mathematics, both for the selection and the treatment of topics. For example, a definition is given for a “positive” real number which reflects the completeness property of the reals rather than the order property, and the section on mathematical induction includes definition by induction as well as proof by induction. The familiar “laws of exponents” are proved as theorems, using an induction argument; and some proofs are to be found in these chapters which make rather novel use of the decimal representation for a real number. Also of interest to the high school teacher is a section on infinite sequences of rational numbers and the chapter on functions.

Tacoma Shuffle

1998 ◽  
Vol 91 (3) ◽  
pp. 212-216
Author(s):  
Lyman S. Holden ◽  
Loyce K. Holden
Keyword(s):  
Problem Solving ◽  
Mathematical Induction ◽  
Computer Application ◽  
Trial And Error ◽  
C Language ◽  
Key Concepts ◽  
Proof By Induction ◽  
Iterative Functions

The key concepts discussed in this article include problem-solving activities, mathematical induction, proof by induction, and use of the phrase “without loss of generality.” Several problem-solving tools are illustrated, such as trial and error, working backward, and seeing patterns. The computer application illustrates recursive and iterative functions using C language.

scholarly journals From Mathematical Induction to Discrete Time

2016 ◽  
Vol 8 (3) ◽  
pp. 75
Author(s):  
David Selke
Keyword(s):  
Discrete Time ◽  
Mathematical Induction ◽  
Cause And Effect ◽  
A Chain ◽  
Logical Sense ◽  
Proof By Induction

Proof by induction involves a chain of implications in which the stages are well ordered. A chain of cause and effect in nature also involves a chain of implications. For this chain to “imply” or bring about its effects in a logical sense, it also has to be organized into a well ordering of stages (which are the points or quanta of time). This means that time must be quantized rather than continuous. An argument from relativity implies that space is quantized as a consequence.

Set representations required for the acquisition of the “natural number” concept

2008 ◽  
Vol 31 (6) ◽  
pp. 655-656 ◽  
Author(s):  
Justin Halberda ◽  
Lisa Feigenson
Keyword(s):  
Natural Number ◽  
Building Blocks ◽  
Number Concept ◽  
Set Representations ◽  
Individual Objects

AbstractRips et al. consider whether representations of individual objects or analog magnitudes are building blocks for the concept natural number. We argue for a third core capacity – the ability to bind representations of individuals into sets. However, even with this addition to the list of starting materials, we agree that a significant acquisition story is needed to capture natural number.

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