Transcript Mathematical Reasoning - New Mexico State University

Lecture 1, MATH 210G.03, Spring 2014
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 (15961650): 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.
Logic: rules of deduction
Logical fallacies I
Fallacies 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
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More fallacies
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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 reelected 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 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.
• study hard.
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informal explanation on origin of WWI
Prior to the First World War, Serbia was part of the AustroHungarian 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. President Dwight 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 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
effect in ecology
Wikipedia: Cascade effect
Deadly domino effect of extinction proved by
boffins
Overfishing Leads to Domino Effect that Can
Harm Ecosystems for Generations, New MetaAnalysis Suggests
• 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 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 first 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 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
Evidence based causality