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Lecture 1, MATH 210G.03, Fall 2013. Reasoning, Part I. Reason is the capacity human beings have to make sense of things, to establish and verify facts , and to change or justify practices, institutions, and beliefs
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Lecture 1, MATH 210G.03, Fall 2013 Reasoning, Part I • Reason is the capacity human beings have to make sense of things, to establish and verify facts, and to change or justify practices, institutions, and beliefs • It is the basis for the ability to make predictions and to influence their own destiny. • The human brain is not good at reasoning
Aristotle (384 BC – 322 BC), Plato's student, defined human beings as rational animals, emphasizing reason as a characteristic of human nature. He defined the highest human happiness or well being (eudaimonia) as a life which is lived consistently, excellently and completely in accordance with reason
Descartes (1596-1650): I am … nothing but a thinking thing; that is a mind, or intellect, or understanding, or reason – words of whose meanings I was previously ignorant .
David Hume (1711-1776): reason is nothing but a wonderful and unintelligible instinct in our souls, which carries us along a certain train of ideas, and endows them with particular qualities, according to their particular situations and relations (consequently, animals can reason)
Kant (1724-1804): it is in fact possible to reason both about the conditions and limits of human knowledge. And so long as these limits are respected, reason can be the vehicle of morality, justice and understanding
Categories of Reason • Cognitive-instrumental reason is the kind of reason employed by the sciences. It is used to observe events, to predict and control outcomes, and to intervene in the world on the basis of its hypotheses; • Moral-practical reason is what we use to deliberate and discuss issues in the moral and political realm, according to universalizable procedures (similar to Kant's categorical imperative); and • Aesthetic reason is typically found in works of art and literature, and encompasses the novel ways of seeing the world and interpreting things that those practices embody.
Logical fallacies IFallacies that arise in Mathematics • Affirming the consequent :”if she runs fast then she must be thin. She is thin. Therefore she runs fast” • Hasty generalization: “Last year I took Logic and I got a C. So did my two friends. It is impossible to get an A in Logic! • Faulty causality : a sequence of events is improperly turned into a causal chain. Example: In November 1989 the Berlin wall came down. In a speech two years earlier, Ronald Reagan challenged Mickhael Gorbachev to “tear down the wall in a speech in Berlin.” (1:50) Therefore, Reagan is responsible for the fall of communism in Europe.
Begging the question or circular reasoning: this type of reasoning does not allow for any kind of progress in the discussion. • "You mean there's a catch?" "Sure there's a catch", Doc Daneeka replied. "Catch-22. Anyone who wants to get out of combat duty isn't really crazy.” • Equivocation (words can have multiple meanings) "Everyone should fight for what they believe in. You disagree with my beliefs, so I'm going to punch you in the nose." • Non sequitur: the conclusion does not follow from the premises. E.g. “Our professor should only give A’s and B’s, as the key to successful learning is participation, and happy students participate more willingly.” (Happy students may be more willing to participate but this consideration is irrelevant when evaluating student work.) • Either-or or false dilemma: improperly limiting to two outcomes when at least another one is possible –popular strategy in political campaigning. ”Either we cut the social programs or we live with a huge deficit and we can't live with the deficit." • Slippery slope: improperly anticipating that one occurrence will unavoidably start a series of other occurrences _______________________________________________________________________________________________________________________________________
More fallacies • Ad hominem: attacks the character of the opponent. "What would Mary know about fixing cars? She is a woman.” • Bandwagon (Ad Populum): the validity and truth of a conclusion is based on a general consensus. E.g., “Everyone’s doing it. Why can’t we” • Appeal to pity/appeal to fear(emotional fallacies) (“you should give me an extension on this paper since my girlfriend left me and someone stole my IPhone!”). “If I am not re-elected then we will lost homeland security” • Appeals to authority. E.g., “God does not exist. Respected scientists like Stephen Hawkins are convinced of this.” • See also “The Crime of Galileo” see also False analogy: This is an analogy drawn from irrelevant or inconsistent elements – it generalizes one non essential similarity between two objects/situations and extends the analogy to other not similar aspects. E.g., "There are seven windows given to animals in the domicile of the head: two nostrils, two eyes, two ears, and a mouth. . . . From this and many other similarities in Nature, too tedious to enumerate, we gather that the number of planets must necessarily be seven."(Francesco Sizzi, 17th-century Italian astronomer) • Intentional and affective ( not logical fallacies per se) : • An intentional fallacy consists of basing the interpretation of a text solely on the “intent or purpose of the author.” E.g., “M.H. Kingston feels that ignoring the exact biographical details relevant to her narrative engages the reader at a deeper level” (What makes it a fallacy, is that you cannot possibly know how M.H. Kingston feels, now or when she was writing…) • Affective fallacy. E.g., “Clearly, …”.
Cause and effect chains • When you study hard you get a good grade • When you get a good grade your allowance goes up • When your allowance goes up you buy a bike • When you buy a bike you ride hard • When you ride hard you win the tour de france • When you win the tour, you get kissed by podium girls. • study hard. • When you study hard you get a good grade • When you get a good grade your allowance goes up • When your allowance goes up you buy a bike • When you buy a bike you crash your bike and break your collarbone. • Don’t study hard.
informal explanation on origin of WWI • Prior to the First World War, Serbia was part of the Austro-Hungarian Empire. This foreign domination was greatly resented. On June 28th, 1914, a nationalist organization called the Black Hand assassinated Archduke Franz Ferdinand in an attempt to assert demands for Serbian independence. It is widely speculated that the Serbian government had no involvement in the assassination. At Germany's insistence Austria-Hungary declared war on Serbia in an effort to crush the nationalist movement. • This event escalated into a global conflict because the major European powers established pre-war alliances in an attempt to avoid war. Britain, France and Russia established the Triple Entente while Germany, Italy and Austria-Hungary formed the Triple Alliance. Such alliances were necessary because each nation had to defend its interests. Britain, for instance, was the undisputed ruler of the seas and feared the expansion of the German navy. France, meanwhile, wanted revenge on Germany for the outcome of the Franco-Prussian war of 1871 • These alliances were meant to prevent war but had the opposite effect. When Austria-Hungary went to war with Serbia, Russia came to defend the Serbs; the Russians and Serbs were naturally aligned because they were both of Slav descent. Germany felt obligated to join the conflict and promptly waged war on Russia. Britain and France had already established an alliance with Russia and had no choice but to wage war on Germany. Within a month of June 28th, 1914, all of the major European powers were at war. Note: Italy held out until 1915 when it joined the allies.
Causality in Math • Causal reasoning is the basis of mathematical deduction • If…then • P: N is an even number. • Q: N is an odd number. • If N is an even number then N+1 is an odd number.
Domino theory/cold war • Referring to communism in Indochina, U.S. PresidentDwight D. Eisenhower put the theory into words during an April 7, 1954 news conference: • Finally, you have broader considerations that might follow what you would call the "falling domino" principle. You have a row of dominoes set up, you knock over the first one, and what will happen to the last one is the certainty that it will go over very quickly. So you could have a beginning of a disintegration that would have the most profound influences.
Domino effect • Seeds of the tree Calvaria major, now found exclusively on the island of Mauritius, must pass through the abrasive gut of a large animal in order to germinate. None of the animals currently on Maritius have that ability. The dodo (25 kg pigeon), hunted to extinction in the late 17th century, probably was the key to recruitment in this species. Only a few very old trees now survive.
Domino theory in the real world is controversial because there is not certainly in the hypotheses • What is the logical process that lets us draw conclusions from assumptions? • Can the logical process be extended to uncertain events?
If the first domino fallsand if whenever a domino falls then the next domino will fall then all of the dominoes will fall.
Domino effect in mathematics Sum of first n equals nth times next after nth over 2 True for n=0 Sum of first k k plus first plus k plus first times next over 2
If the first domino fallsand if whenever a domino falls then the next domino will fall then all of the dominoes will fall.
Domino theory in mathematics The mathematical equivalent of the domino theory is called the “principle of mathematical induction.” Induction: inference of a generalized conclusion from particular instances There is no controversy with the domino theory in mathematics but the induction hypothesis has to be verified.ofa generalized conclusion from particular instances
Clicker question: A dodo is: A) A 25 kg pigeon that was hunted to extinction B) A person who is hopelessly behind the times C) A member of a guitar and drum band called “the dodos” D) All of the above
