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Kernel: SageMath 9.8

Bertrand Russell

from urllib.request import urlopen
from IPython.display import Image
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Biography

  • Bertrand Russell, a British philosopher, logician, and social reformer, was born on May 18, 1872, in Treelike, Monmouthshire, Wales, and passed away on February 2, 1970, in Penrhyndeudraeth, Merioneth.

  • He played a pivotal role in the analytic movement in Anglo-American philosophy and was awarded the Nobel Prize for Literature in 1950. Russell's significant contributions to logic, epistemology, and the philosophy of mathematics cemented his position as one of the leading philosophers of the 20th century.

  • Russell attended Trinity College Cambridge, where he studied mathematics. Later he returned to Cambridge as a lecturer and then as a fellow at Trinity College. Russell's association with Cambridge continued throughout his life, and he made significant contributions to the university's intellectual environment. Later in life he taught at a number of universities including: The University of London, Harvard, City College of New York, and UCLA. His presence at CCNY at UCLA sparked controversy among conservative groups, who objected to his political views and pacifism.

  • Beyond academia, he was renowned for his advocacy for peace and his writings on social, political, and moral issues. Throughout his life, Russell authored over 70 books and approximately 2,000 articles, was married four times, engaged in numerous public controversies, and received both admiration and criticism worldwide.Britannica

  • Notably, Russell's Doctoral advisor Alfred North Whitehead was a man of equal renown and co-authored with Russell "Principia Mathematica", (not to be confused with Newton's Principia). The primary aim of this monumental work was to provide a rigorous and formal foundation for mathematics based on logic.Stanford

  • Russell was as much of a renaissance man as anyone could have to aspired to be. In his autobiography he speaks of three passions, "simple but overwhelmingly strong", that governed his life: the longing for love, the search for knowledge, and an unbearable pity for the suffering of mankind. His intellect, like all of our own, was colored by his nature; which is just as responsible for his brilliance as his propensity towards mathematics.

Contributions to Discrete Mathematics

Russell's Paradox

To begin understanding Russell's paradox, the definition of "Naïve Set Theory" may be Appropriate. Naive set theory is an elementary approach to understanding sets within mathematics, typically characterized by the unrestricted comprehension axiom, which allows for the formation of sets based on any property. Russell's paradox is a foundational problem in set theory that highlights the inherent contradictions within naive set theory. The paradox revolves around the set of all sets that do not contain themselves.

R = {x | x ∉ x}

If such a set exists, it leads to a logical contradiction: If it contains itself, it shouldn't belong to the set; if it doesn't contain itself, then it should belong to the set. This paradox challenges the notion of unrestricted comprehension, which allows the formation of sets based on any property. Russell's discovery shook the foundations of set theory and underscored the need for a more rigorous axiomatic approach to mathematics, leading to the development of modern set theory and the introduction of principles like the axiom of regularity to resolve such contradictions.Stanford

Theory of Types

Russell's paradox arises from the assumption that any property can define a set, leading to contradictions such as the set of all sets that do not contain themselves. Georg Cantor, the founder of set theory, recognized early on the dangers of unrestricted set formation and distinguished between sets and too-large collections called classes to avoid inconsistencies. Russell's response was the theory of types, which prevents self-referential set definitions by organizing propositional functions into a hierarchy, ensuring that sets and conditions applied to them are of the same "type." This approach is underpinned by the vicious circle principle, which mandates specifying the domain of a function before its definition, aiming to restore consistency to mathematics by avoiding self-application and the paradoxes it creates.

Principia Mathematica

"Principia Mathematica," authored by Alfred North Whitehead and Bertrand Russell, stands as a seminal work in the realm of formal logic. This monumental undertaking was born from Russell's intent to formalize his logicism program initially outlined in "The Principles of Mathematics." Serving as a defense of logicism, the thesis that mathematics is reducible to logic, "Principia Mathematica" wielded significant influence in shaping modern mathematical logic and sparked extensive research into the foundations of mathematics in the 20th century. Unlike Frege's system, "Principia Mathematica" introduced the theory of types to circumvent Russell's Paradox, emphasizing a logic of relations over functions and objects. Its formulation of sets using higher-order functions addressed foundational issues within set theory, while its construction of natural numbers diverged from Frege's approach, employing the axiom of infinity and a theory of ramified types. The completion of "Principia Mathematica" raised philosophical questions about the additional axioms it relied upon, stimulating debate over their status as logical truths. Despite its departure from contemporary set theory's arithmetization of analysis, "Principia Mathematica" remains a cornerstone in the study of mathematical logic, offering unique insights into mathematical concepts and their formalization.Stanford

Reflections

Nature of the Information

Considering the fact that Bertrand Russell is more or less contemporaneous, relative to the timeline that envelops the mathematical endeavor, the information gathered was easily accessible, digestible and overall accurate. All things considered Russell rests in a sweet spot, where, he is modern enough so that his story and pursuits are far from enigmatic, but no so modern that his perception is subject to the scrutiny of sites like Wikipedia. Perhaps it is because the man is genuinely worthy of reverence, but there is nothing but consistency in his praise and written work.

Contribution

Regarding my own contribution, it can feel overwhelming at times to try and grasp the immensity that is the digital world. Decades prior to my arrival on earth, the groundwork was already laid for humanity's most recent monumental achievement. If I am to reflect, genuinely, on the possibility of my impact, it will be in an attempt to change the nature of it. Through the course of this project the best parts of the internet have made itself plain, i.e., that it is a repository for the vast storehouse that is the collective knowledge of humanity; however, this is just one edge to the sword. Certainly because of my generational proximity I have made some pretty useless contributions such as facebook, twitter, and instagram posts, but hopefully in the coming years the profound will outweigh the mundane.

References