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Ashlin MeuserPHIL 1154Professor Fosil 3-21-19Russell Bertrand And The Pursuit Of truthThe late 19th into the 20th century was a time of great change in the world of philosophy, mathematics, and logic but with this great change came great struggles. At the forefront of these changes was Bertrand Russell, a man set on discovering absolute truth. From an early age Russell was infatuated with Mathematics, mainly because it contains absolute truth (or so he thought). It was when Russell learned of axioms that he realized even mathematics did contain absolute truth. An axiom is a mathematical statement or property considered to be self-evidently true, but yet cannot be proven. To Russell this spelled disaster for his quest for truth. It was only when he was introduced to logic that he realized logic is the foundation of all mathematics. Bertrand Russell published his first findings on the subject in 1903. The text, entitled “Principia Mathematica” contained not only his theory that logic is the foundation of Mathematics but also the famed paradox known as Russell’s Paradox. With every new discovery came disruption among other philosophers and mathematicians who had not yet realized these advancements.At the time many prominent philosophers and mathematicians found inspiration in the work of other like-minded individuals. To the young Bertrand Russell, none had such a lasting impact than Gottlob Frege. Gottlob Frege was German mathematician who had written the enigmatic German text called “Bergriffsschrift”(Doxiadis 119). It introduced “concept script” a fully logical language similar to Leibniz’s language. Frege argued that logic was meant to model reality not to be used as a calculating tool. This a different view of logic compared to the better-known philosophers at the time that saw logic as a way to calculate. To Friege the ordinary language was not “suited to science”(Doxiadis 122). The ordinary language used in logic was viewed by many philosophers and Mathematicians of the period as far too simple for mathematics. This led to new ideas of logic in order to meet the needs of mathematicians like Russell attempting to study logic from a mathematical viewpoint. The logic of Aristotle and Boole lacked variables, which makes things difficult when trying to understand the mathematical roots of logic. In order to solve this problem Frege invented a, Notation of quantifiers?and variables. (In natural language, generality is represented by inserting an expression like “everything” or “something” in the argument-place of the?predicate; in the notation used in logic since Frege, the argument-place is filled by a?variable?letter, say?x, and the resulting expression prefixed by a quantifier, “For every?x” or “For some?x,” said to “bind” that variable (Baldwin).During the second volumes printing of “Grundgesetze” in 1902 Frege received a letter from Russell informing him that he had discovered a paradox that effectively disproved all of volume one and two of “Grundgesetze”.Russell’s paradox published in 1903 demonstrated a limitation of the system that Frege had created. The system created by Frege can be described using sets and “set builder notation”. The paradox is explained as the set of all sets that are not members of themselves. The set appears as a member of itself if and only if it is not a member of itself. For example, the set of all knives, are not members of themselves. Other sets, such as the set of all non-Knives, are members of themselves. The set of all sets that are not members of themselves can be called “K”. If “K” is a member of itself, then it must not be a member of itself and the same follows for if “K” is not a member of itself. Russell discovered this paradox after he applied the same kind of reasoning used in Cantor’s diagonal argument to a hypothetical class of imaginable objects. In Russell’s “Introduction to Mathematical Philosophy” he describes his paradox:The comprehensive class we are considering, which is to embrace everything, must embrace itself as one of its members. In other words, if there is such a thing as “everything,” then, “everything” is something, and is a member of the class “everything.” But normally a class is not a member of itself. Mankind, for example, is not a man. Form now the assemblage of all classes which are not members of themselves. This is a class: is it a member of itself or not? If it is, it is one of those classes that are not members of themselves, i.e., it is not a member of itself. If it is not, it is not one of those classes that are not members of themselves, i.e. it is a member of itself. Thus of the two hypotheses – that it is, and that it is not, a member of itself – each implies its contradictory. This is a contradiction (Bertrand 1919, 136).After the discovery of the paradox in 1901 Russell teamed up with Alfred Whitehead a fellow Mathematician to write “Principia Mathematica” which would seek to “fix” the problem of the faulty system that Frege had created. Russell created a theory called “set language” which stated that, “a set of one type can only include sets of a lower” (Doxiadis 175). This eliminates the paradox and allows the system that Frege created to work. The intention of “Principia Mathematica” was to not only fix the system of interpretation that Ferge had formalized for set theory but also to simplify it for the more general reader. To do this the duo set out to prove that 1+1=2 with the use of logic. Russell and Whitehead encountered numerous limits to their new version of Type Theory. The first of many was how to simplify something so complicated into something a child could understand. This led to years in writing and revising of the premises because Russell discovered that his premises were far too complicated for the resulting simple argument. To combat this problem a new set of axioms were used. With the revised axioms and premises Russell and whitehead were finally able to prove 1+1=2. Their findings led to other advancements in the field.Kurt G?del an Austrian logician used Freges “concept theory” language to analyze and understand the foundations of mathematics. He published two Incompleteness Theorems in 1931. These theorems helped show the limitations of every axiomatic system that could be used for arithmetic. This was an important milestone in proving that logic was the foundation of mathematics. It proved that while a complete set of axioms for mathematics has not been found they do exist. The limitations of formal systems had not yet fully been realized until G?del’s theorems. While studying “Principia” G?del assigned each logical symbol a unique number called G?del numbers. He then assigned a unique number formed by multiplying the G?del numbers to represent each statement created with the logical symbols. This brought him to the realization that “Principia Mathematica” was a book based in numbers, which used these very numbers to prove itself. G?del’s original goal after doing this was to find inconsistency but what he discovered was much more intriguing. G?del created a theorem that was contained within “Principia Mathematica” that was impossible to prove within the system. “This formula is unprovable by the rules of “Principia Mathematica” (G?del) but this could exist within Russell and Whiteheads system thus creating a paradox. With this discovery G?del concluded that there was at least one undecidable proposition and is therefore incomplete. With that the “Incompleteness Theory” was born.The most famous philosophers, logicians, and mathematicians of the 19th and 20th century all encountered limits within logic. Despite these limitations they were able to prove some of the most important theories of their time. The dedication to truth and the pursuit of it has led to some of the most crucial advancements in our society. The computer and computer science would not be possible had logic not been re-invented during the late 19th and early 20th centuries. Russell Bertrand was at the forefront of quest for absolute truth. Without his work we might not have won WWII or had been able to put a man on the moon. His and the other logician’s work shall live on forever.Works CitedBaldwin, John T. “What Is Russell's Paradox?”?Scientific American, article/what-is-russells-paradox/.?Doxiádis, Apóstolos K., et al.?Logicomix. La Librairie Vuibert, 2018.?G?del's Proof, Nagel & Newmann, New York University Press, 2001.Russell, Bertrand.?Introduction to Mathematical Philosophy. New York Press, 1919. ................
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