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Encyclopedia > Principia Mathematica
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The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. It is an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic. One of the main inspirations and motivations for the Principia was Frege's earlier work on logic, which had led to some contradictions discovered by Russell. These were avoided in the Principia by building an elaborate system of types: a set has a higher type than its elements and one can not speak of the "set of all sets" and similar constructs which lead to paradoxes (see Russell's paradox). Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Jump to: navigation, search Alfred North Whitehead, OM (February 15, 1861, Ramsgate, Kent, UK – December 30, 1947, Cambridge, MA) was a British philosopher, physicist, and mathematician who worked in logic, mathematics, philosophy of science and metaphysics. ... Jump to: navigation, search Bertrand Russell The Right Honourable Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was an influential British logician, philosopher, and mathematician, working mostly in the 20th century. ... Jump to: navigation, search 1910 was a common year starting on Saturday (see link for calendar). ... Jump to: navigation, search 1913 is a common year starting on Wednesday. ... Jump to: navigation, search In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 – July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory. ...


The Principia only covered set theory, cardinal numbers, ordinal numbers and real numbers; deeper theorems from real analysis were not included, but by the end of the third volume it was clear that all known mathematics could in principle be developed in the adopted formalism. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Alternative meaning: number of pitch classes in a set. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... Please refer to Real vs. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...


The questions remained whether a contradiction could be derived from the Principia's axioms, and whether there exists a mathematical statement which could neither be proven nor disproven in the system. These questions were settled by Gödel's incompleteness theorem in 1931. Gödel's second incompleteness theorem shows that basic arithmetic cannot be used to prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger. In other words, the statement "there are no contradictions in the Principia system" cannot be proven true or false in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false). Jump to: navigation, search In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ... Jump to: navigation, search 1931 is a common year starting on Thursday. ...


Yet, as Douglas Hofstadter [[1]] has pointed out, there may be additional levels of potential contradiction here. A central principle of the "system of types" mentioned above is that statements that are self referential are forbidden, to avoid Russell's paradox. Loops of statements that are self referential (circular definitions) are also forbidden. However, the statement "We do not allow self-referential statements in Principia Mathematica" is a violation of the rule against self-referential statements, an apparant contradiction at the heart of the philosophy. Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory. ...


A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third. Jump to: navigation, search Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ...


The Principia is widely considered by specialists in the subject to be one of the most important and seminal works in mathematical logic and philosophy.


Quote from the book

  • "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." – page 362.

See also

Begriffsschrift is the name of a book on logic by Gottlob Frege published in 1879. ...

External links


  Results from FactBites:
 
Philosophiae Naturalis Principia Mathematica - Wikipedia, the free encyclopedia (2653 words)
The Philosophiae Naturalis Principia Mathematica (Latin: "mathematical principles of natural philosophy", often Principia or Principia Mathematica for short) is a three-volume work by Isaac Newton published on July 5, 1687.
Descartes' book of 1644 Principia philosophiae (Principles of philosophy) stated that bodies can act on each other only through contact: a principle that induced people, among them he himself, to conjecture an universal medium as the carrier of interactions such as light and gravity— the aether.
It was perhaps the force of the Principia, which explained so many different things about the natural world with such economy, that caused this method to become synonymous with physics, even as it is practised almost three and a half centuries after his beginning.
  More results at FactBites »

 
 

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