Principia Mathematica
See: Bertrand Russell, Book, Axiom, Science, Philosophiæ Naturalis Principia Mathematica, Alfred North Whitehead, Gödel's Incompleteness Theorem, Russell's Paradox, System of Types.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Principia_Mathematica Retrieved:2015-8-23.
- 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, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced '✸9 and an all-new Appendix C.
PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.
One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.
PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: "The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions."
The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.
- 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, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced '✸9 and an all-new Appendix C.
2014
- (Avigad & Harrison, 2014) ⇒ Jeremy Avigad, and John Harrison. (2014). “Formally Verified Mathematics.” In: Communications of the ACM Journal, 57(4).In: Communications of the ACM Journal, 57(4). doi:10.1145/2591012
- QUOTE: From the point of view of the foundations of mathematics, one of the most significant advances in mathematical logic around the turn of the 20th century was the realization that ordinary mathematical arguments can be represented in formal axiomatic systems in such a way their correctness can be verified mechanically, at least in principle. Gottlob Frege presented such a formal system in the first volume of his Grundgesetze der Arithmetik, published in 1893, though in 1903 Bertrand Russell showed the system to be inconsistent. Subsequent foundational systems include the ramified type theory of Russell and Alfred North Whitehead's Principia Mathematica, published in three volumes from 1910 to 1913; Ernst Zermelo's axiomatic set theory of 1908, later extended by Abraham Fraenkel; and Alonzo Church's simple type theory of 1940. When Kurt Gödel presented his celebrated incompleteness theorems in 1931, he began with the following assessment:
1927
- (Whitehead & Russell, 1927) ⇒ Alfred North Whitehead, and Bertrand Russell. (1927). “Principia Mathematica, 3 vols, 2nd ed." Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols 2, 3). Abridged as Principia Mathematica to *56, Cambridge University Press, 1962.