Lecture 0, MATH 210G.02, Fall 2016 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 Rules of logic: allow to draw conclusions with same degree of certainty as supporting hypotheses
Pliny the Eldar, Naturalis Historia, Book II Section 7: The only certainty is that nothing is certain. Benjamin Franklin: In this world nothing can be said to be certain, except death and taxes.
Math and Reason Math involves symbolic representation of quantities and manipulation of those quantities. The symbols and rules for manipulating the symbols conform to logical rules. Encoding quantities in symbols enables one to avoid confusing language and logic and enables one to establish mathematical truths once and for all. To motivate symbolic manipulation of quantities we want to exam some of the sorts of logical traps that are easy to fall into when distracted by context. First, here is a brief history of thought on human 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, jus tice 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.
Logic: rules of deduction
Logical fallacies I Fallacies that arise in Mathematics Example 1: Afrming the consequent : if she runs fast then she must be thin. She is thin. Therefore she runs fast
Afrming the consequent: If I am in T or C then I am in NM. I am in NM. Therefore, I am in T or C.
What is the fallacy: If he runs fast then he must be thin. He runs fast. Therefore he is thin.
Example 2: 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! Another example: Every even number is the sum of two prime numbers: 2=1+1; 20=9+11; 100=53+47 etc.
Example 3: 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.
Other logical fallacies (that may or may not arise in math) Non-sequiter Begging the question False dilemma Equivocation Argumentum ad hominum Argumentum ad populem Argumentum ad metum Intentional fallacy Association fallacy Appeal to authority
Non-sequiter: Does not follow
Begging the question: "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.
False dilemma: there is only one choice
Equivocation:
Argumentum ad hominem:
Argumentum ad populem:
Argumentum ad metum: appeal to fear
Intentional fallacy:
Association fallacy:
Appeal to authority:
Confrmation bias: Selective attention :
Causality revisited Mathematical deduction starts with one or more hypotheses, combines with rules of logic, and draws conclusions that have certainty equal to that of the hypotheses In real life, there is some uncertainty in each of the deductive steps. This is the basis of the slippery slope. Sometimes the slippery slope is a valid concern but ofen is not.
Slippery slope: the frst step to becoming Amy Winehouse
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 crash your bike and break your collarbone. Don t 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 ride hard When you ride hard you win the tour de france When you win the tour, you get kissed by podium girls.
informal explanation on origin of WWI Prior to the First World War, Serbia was part of the AustroHungarian Empire. 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. Major European powers established pre-war alliances to avoid war by defending its own interests. Alliances had the opposite effect. When Austria-Hungary went to war with Serbia, Russia came to defend the Serbs( both Slavic) Germany felt obligated wage war on Russia. Britain and France, allied with Russia had no choice but to wage war on Germany. Within one month, all of the major European powers were at war.
Causality in Math Causal reasoning is the basis of mathematical deduction If then P: N is an even number. Q: N+1 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. President Dwight D. Eisenhower put the theory into words on April 7, 1954: 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 frst one, and what will happen to the last one is the
Domino effect in Ecology 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.
Links to other evidence of domino (cascade) effect in ecology Wikipedia: Cascade effect Deadly domino effect of extinction proved by bofn Overfshing Leads to Domino Effect that Can Harm
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 frst domino falls and 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 plus k plus first k plus first times next over 2
If the frst domino falls and 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 verifed.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