Introduction to Mathematical PhilosophyNot to be confused with the philosophy of mathematics, mathematical philosophy is the structured set of rules that govern all existence. Or, in a word: logic. While this branch of philosophy threatens to be an intimidating and abstract subject, it is one that is surprisingly simple and necessarily sensible, particularly at the pen of writer Bertrand Russell, who infuses this work, first published in 1919, with a palpable and genuine desire to assist the reader in understanding the principles he illustrates. Anyone interested in logic and its development and application here will find a comprehensive and accessible account of mathematical philosophy, from the idea of what numbers actually are, through the principles of order, limits, and deduction, and on to infinity. British philosopher and mathematician BERTRAND ARTHUR WILLIAM RUSSELL (1872-1970) won the Nobel Prize for Literature in 1950. Among his many works are Why I Am Not a Christian (1927), Power: A New Social Analysis (1938), and My Philosophical Development (1959). |
Contents
| 1 | |
| 7 | |
FINITUDE AND MATHEMATICAL INDUCTION | 20 |
THE DEFINITION OF ORDER | 29 |
KINDS OF RELATIONS | 42 |
SIMILARITY OF RELATIONS | 52 |
RATIONAL REAL AND COMPLEX NUMBERS | 63 |
INFINITE CARDINAL NUMBERS | 77 |
LIMITS AND CONTINUITY OF FUNCTIONS | 107 |
SELECTIONS AND THE MULTIPLICATIVE AXIOM | 117 |
THE AXIOM OF INFINITY AND LOGICAL TYPES | 131 |
INCOMPATIBILITY AND THE THEORY OF DEDUCTION | 144 |
PROPOSITIONAL FUNCTIONS | 155 |
DESCRIPTIONS | 167 |
CLASSES | 181 |
MATHEMATICS AND LOGIC | 194 |
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Common terms and phrases
a-functions aliorelative argument arithmetic assert assume asymmetrical asymmetrical relation author of Waverley axiom of infinity belongs called Cantor cardinal number chapter classes of classes commutative law complex numbers consists converse domain correlation Dedekindian deduction defined example existence fact finite follows formally equivalent fractions generalised given identical inductive cardinal inductive numbers inference infinite number integers irrational less limit limiting-points logical logical constants mathematical induction means multiplicative axiom namely natural numbers notion null-class number of individuals number of terms object one-many relations one-one relation ordinal Peano's philosophy of mathematics possible posterity premisses primitive ideas primitive propositions Principia Mathematica progression propositional function prove real numbers reflexive relation-numbers sense serial number series of ratios set of terms similar so-and-so Socrates sometimes true square sub-classes successor suppose symbols theory thing tion truth-functions unicorn upper section values variable words
Popular passages
Page 3 - The most obvious and easy things in mathematics are not those that come logically at the beginning; they are things that, from the point of view of logical deduction, come somewhere in the middle. Just as the easiest bodies to see are those that are neither very near nor very far, neither very small nor very great, so the easiest conceptions to grasp are those that are neither very complex nor very simple (using "simple
Page 4 - It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number 2: the degree of abstraction involved is far from easy.