# Reflection Paper on a Beautiful Mind

A Beautiful Mind is a story based on the life of the famous mathematician John Forbes Nash, Jr. His contributions to mathematics are outstanding. When he was an undergraduate, he proved Brouwer’s fixed point theorem.

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He then broke one of Riemann’s most perplexing mathematical problems and became famous for the Nash Solution. Game Theory from then on, Nash provided breakthrough after breakthrough in mathematics. In 1958 John Forbes Nash was described as being ‘the most promising young mathematician in the world’. John solved problems in mathematics that many mathematicians deemed not solvable.

On the threshold of such a promising and outstanding career, he then went on to suffer through three decades of a devastating form of paranoid schizophrenia. He lost his teaching professions and his job. He refused all medical treatment and spent years in and out of delusional states. Remarkably, in 1994 John won the Nobel Prize in Economic Science for his work on Game Theory, he was only 21 when he wrote his paper. The story of John Forbes Nash Jr. is definitely connected with Logic, as he is a mathematician. In his breaking of many perplexing mathematical problems, he used logic.

And in producing his famous Game theory, logical systems were used. Among the important properties that logical systems can have: ?Consistency, which means that no theorem of the system contradicts another. ?Validity, which means that the system’s rules of proof will never allow a false inference from true premises. A logical system has the property of soundness when the logical system has the property of validity and only uses premises that prove true (or, in the case of axioms, are true by definition). ?Completeness, which means that if a theorem is true, it can be proven. Soundness, which means that the premises are true and the argument is valid. In one of the scenes in the movie, where Nash was in a bar with friends, the use of logic was very evident. While Nash and his friends all have their eyes on the same Blonde woman, you begin to sense his ‘genius moment’. He surprises his colleagues with the question that if we all want the same woman, nobody wins, if we all go after her friends, nobody wins, and thus there must be a solution to ensure that everyone wins. With that, Nash writes a formula on a napkin, rushes out of the bar and works feverishly on his new theory.

The scene shows the art of logic, which examines three acts of the reason: simple apprehension, judgment and reasoning. Simple Apprehension is the grasp of a concept. A concept is also called an idea, a species, an intelligible form, and a mental word. A concept has an extension, which is the group of things included under the concept. A singular concept is the concept of one individual, for example, your concept of President Abraham Lincoln. A universal concept extends to a whole class of things. The more features (or notes) included in a universal concept, the narrower its extension.

A transcendental concept is one that applies to anything that exists, for example being, thing, unit, distinct, good, true, beautiful. A concept by itself is not true or false, just as a single word is not true or false. A concept is a sign of a thing, just as a spoken word is a sign of a concept, and through the concept the spoken word is a sign of the thing. A concept is not merely a sensation stored in the mind, but is an immaterial act of understanding. A Judgment is expressed in a complete sentence or proposition.

Judgments are either attributive, when we say “A is B”, where A is a subject and B is a predicate, or existential, as when we say “A exists”. Affirmation or affirmative judgment is called composition, because we are putting two concepts together. Negation is called division, because we are taking two concepts apart. A judgment is either true or false. Reasoning involves three terms or concepts, and two judgments. The major term is the broadest, the minor term is the narrowest, and the middle term is between the two, included in the meaning of the major term, and including in it the meaning of the minor term.

An example is given in the table below (The Structure of a Syllogism). Major PremiseEvery man is mortal. Mortal = major term Minor PremiseKenji is a man. Man = middle term ConclusionKenji is mortal. Kenji = minor term A syllogism is the verbal expression of an act of reasoning. In a syllogism or perfect argumentation, where one thing is given, another thing necessarily follows. Other forms of argumentation give lesser degrees of certainty. A perfect syllogism employes deduction, which is reasoning that starts from general truths, and then applies them in a particular instance.