Lecture 1, MATH 210G.03, Spring 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
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
•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.
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.
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.”
Equivocation:
Non-sequiter: Does not follow
False dilemma: there is only one
choice
Slippery slope: the frst step to
becoming Amy Winehouse
Argumentum
ad hominem:
Argumentum ad
populem:
Argumentum ad
metum: appeal
to fear
Intentional fallacy:
Association
fallacy:
Appeal to
authority:
Confrmation
bias:
Selective attention
:
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.
•informal explanation on origin of WWI
•Prior to the First World War, Serbia was part of the Austro-Hungarian
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 certainty that
it will go over very quickly.
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