## Generalized Catalan Numbers

*Thomas Koshy*

### in Catalan Numbers with Applications

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0017
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter generalizes Catalan numbers and introduces Fuss numbers.

## Divisibility Properties

*Thomas Koshy*

### in Catalan Numbers with Applications

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0013
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates several divisibility properties of shows Catalan numbers. They include their parity and primality. Mersenne numbers are also described.

## A Catalan Triangle

*Thomas Koshy*

### in Catalan Numbers with Applications

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0014
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates an interesting triangular array that contains Catalan numbers and Cayley numbers.

## Partitions and Catalan Numbers

*Thomas Koshy*

### in Catalan Numbers with Applications

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0010
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter investigates several occurrences of Catalan numbers in the theory of partitioning. Noncrossing partition is explained in detail.

## A Family of Binary Words

*Thomas Koshy*

### in Catalan Numbers with Applications

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0015
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter generalizes the family of binary numbers studied in Chapter 5.

## Notes on memory (1945)

*Alan M. Turing*

### in Alan Turing's Automatic Computing Engine: The Master Codebreaker's Struggle to build the Modern Computer

- Published in print:
- 2005
- Published Online:
- January 2008
- ISBN:
- 9780198565932
- eISBN:
- 9780191714016
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198565932.003.0022
- Subject:
- Mathematics, History of Mathematics

This chapter presents fragments from Turing's draft of ‘Proposed Electronic Calculator’. This material is of interest chiefly because of its remarks concerning the universal machine of Turing's 1936 ... More

## Science in the Looking Glass: What Do Scientists Really Know?

*E. Brian Davies*

- Published in print:
- 2007
- Published Online:
- September 2008
- ISBN:
- 9780199219186
- eISBN:
- 9780191711695
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219186.001.0001
- Subject:
- Physics, History of Physics

How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book ... More

## Catalan Numbers with Applications

*Thomas Koshy*

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

Fibonacci and Lucas sequences are “two shining stars in the vast array of integer sequences,” and because of their ubiquitousness, tendency to appear in quite unexpected and unrelated places, ... More

## In the Wilderness: The Doctrine of Defilement in the Book of Numbers

*Mary Douglas*

- Published in print:
- 2001
- Published Online:
- October 2011
- ISBN:
- 9780199245413
- eISBN:
- 9780191697463
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199245413.001.0001
- Subject:
- Religion, Biblical Studies

This book is a classic work on the anthropology of the Book of Numbers. Up to now Bible scholars have tended to dismiss Numbers — Wellhausen called it the junk room of the Bible, and most ... More

## General Theorems From Specific Images

*Marcus Giaquinto*

### in Visual Thinking in Mathematics

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199285945
- eISBN:
- 9780191713811
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285945.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This chapter shows that some general arithmetical theorems almost certainly can be discovered by visualizing. First, it describes two examples of what look like acceptable routes to general theorems ... More

## Binomial Coefficients

*Thomas Koshy*

### in Catalan Numbers with Applications

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents a brief introduction to binomial coefficients, the cornerstone of all the discussions in the book. Among the properties discussed are Hermite's divisibility properties. Catalan ... More

## The Central Binomial Coefficient Revisited

*Thomas Koshy*

### in Catalan Numbers with Applications

- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0003
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter continues to investigate the central binomial coefficient in much greater detail. Topics explained include Wallis' product, Lucas numbers, Fermat's little theorem, Carlitz's formula, ... More

## Letters-to-Numbers Problems

*Patrick R. Laughlin*

### in Group Problem Solving

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691147918
- eISBN:
- 9781400836673
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147918.003.0006
- Subject:
- Psychology, Social Psychology

This chapter looks at letters-to-numbers problems, an interesting class of problems that entail many insightful strategies. Letters-to-numbers problems strongly fulfill the four conditions of ... More

## On Measurement and Numbering

*Rory Fox*

### in Time and Eternity in Mid-Thirteenth-Century Thought

- Published in print:
- 2006
- Published Online:
- May 2006
- ISBN:
- 9780199285754
- eISBN:
- 9780191603563
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199285756.003.0007
- Subject:
- Philosophy, Philosophy of Religion

This chapter examines 13th century views on issues of measurement, particularly temporal measurement. It shows that 13th century thinkers distinguished between a variety of ways in which the word ... More

## The Concept of an Object in Formal Ontology

*E. J. Lowe*

### in The Four-Category Ontology: A Metaphysical Foundation for Natural Science

- Published in print:
- 2005
- Published Online:
- May 2006
- ISBN:
- 9780199254392
- eISBN:
- 9780191603600
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199254397.003.0005
- Subject:
- Philosophy, Metaphysics/Epistemology

The formal ontological concept of an object is explicated and contrasted with that of a property. F. P. Ramsey’s objections to this distinction are challenged. The sense in which objects possess an ... More

## The Mathematics of Various Entertaining Subjects: Research in Recreational Math

*Jennifer Beineke and Jason Rosenhouse (eds)*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164038
- eISBN:
- 9781400881338
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164038.001.0001
- Subject:
- Mathematics, History of Mathematics

The history of mathematics is filled with major breakthroughs resulting from solutions to recreational problems. Problems of interest to gamblers led to the modern theory of probability, for example, ... More

## Frege's Account of Classes

*Michael Potter*

### in Reason's Nearest Kin: Philosophies of Arithmetic from Kant to Carnap

- Published in print:
- 2002
- Published Online:
- May 2007
- ISBN:
- 9780199252619
- eISBN:
- 9780191712647
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199252619.003.0005
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

In the Grundlagen, Frege sketched a proof of the basic properties of natural numbers from the numerical equivalence and showed how that equivalence could be proved if numbers were explicitly defined ... More

## Mathematical Knowledge and the Interplay of Practices

*José Ferreirós*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction ... More

## The Institutions of Democracy

*André Béteille*

### in Democracy and Its Institutions

- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780198080961
- eISBN:
- 9780199082049
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198080961.003.0001
- Subject:
- Sociology, Politics, Social Movements and Social Change

This chapter examines the ideals of democracy and the actual operation of its institutions in India, focusing on those political institutions that began to emerge from the middle of the nineteenth ... More

## Deductive Narrative and the Epistemological Function of Belief in Mathematics: On Bombelli and Imaginary Numbers

*Federica La Nave*

### in Circles Disturbed: The Interplay of Mathematics and Narrative

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691149042
- eISBN:
- 9781400842681
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149042.003.0003
- Subject:
- Mathematics, History of Mathematics

This chapter explores the epistemological function of belief in mathematical thinking by considering Rafael Bombelli's contribution to the creation of imaginary numbers. The discussion focuses on ... More