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PROFESSOR: Hi, welcome
to Calculus Part 2,
where our theme for
the entire course
will essentially be functions
of several variables.
But a more underlying theme,
a theme which will not only
permeate this
course, but virtually
every course in the mathematics
curriculum, and perhaps
other curricula as
well, is the idea
of what we mean by a
mathematical structure.
And if this sounds a little
bit ominous and frightening,
the idea of a
mathematical structure
can be compared very
nicely in terms of a game,
and for this reason,
I have chosen
to entitle our first lesson
"The 'Game' of Mathematics".
"The 'Game' of Mathematics."
And let's emphasize
the word "game" here.
We do not mean game
in a trivial sense,
where in elementary school, the
first row races the second row
to see who finishes the
addition problem first.
The idea that we want to
talk about is what is a game.
And I remember an
old riddle when
I was in about the fourth or
fifth grade, when somebody said
to me: "what is it that looks
like a box, smells like cheese,
and flies?"
And the answer was
a flying cheese box.
And the interesting
point is that this
is how, in the scientific world,
we often make up definitions.
Namely, to define a
game, we try to think
of every single ingredient
that is common to every game,
and then we roll all
of these ingredients
into one long
definition, and that
becomes our final definition.
And with that in mind, let
me take the following tack.
And again, let me point out that
I am going through this rather
hurriedly because the
main aim of the lesson
is to give an
overview, with the idea
that the supplementary notes
and the exercises in the unit
will give you the computational
drill that you need.
But for a first approximation,
let us say that, in any game,
you must have definitions, so to
know what terminology you have.
Then you must have
rules of the game,
and notice that the
rules of the game
are relationships
between the terms.
Oh, as a trivial example,
in playing cards,
there are many
different card games
that can be played with
the same deck of cards.
What makes the game
different are not
the definitions involved,
but the rules of the game.
And finally, there
are objectives.
Well, obviously, the objective
of any game is to win.
What we really mean
by an objective
is the art of carrying
out a winning situation
by successfully employing the
definitions and the rules.
And the way we do that is
usually called strategy.
Strategy is the art of using
the definitions and rules
to carry out the objective
in an inescapable manner.
And the reason I like this
particular little setup
is it gives me a
way to show you,
in juxtaposition, the role
of rote verses reason, logic
verses memory, in any
mathematical situation
or real-life situation.
Namely, the strategy
part of a game is logic,
and things like the
definitions and the rules
are things that we memorize.
OK, now, the thing
that we're going
to do, in our particular
course, is always come back
to this particular structure.
In other words, our
definition of a game
is any system which consists
of definitions, rules,
and objectives, where the
objective is carried out
as an inescapable consequence
of the definitions and the rules
by means of strategy.
And by the way, don't
take this lightly.
This is a very serious topic.
Later in the course,
the computation
will become
sufficiently difficult
that we may lose sight of the
forest because of the trees.
Right now, what I want to
do is emphasize this to you
in terms of topics that
you're already familiar with,
so that you can see what
the overall structure
of mathematics is,
so that you will not
be preoccupied with
this when we're learning
more computational things.
Let me just take
a look at, well,
a relatively trivial example.
I call this a new
look at counting.
We all know how to count.
We start with a number
called 1, and we
have various definitions, 1
plus 1 is 2, 2 plus 1 is 3,
3 plus 1 is 4, 4 plus 1 is
5, 5 plus 1 is 6, 6 plus 1
is 7, et cetera.
OK, so far, so good.
Now, we ask the question:
"how much is 4 plus 3?"
Now, obviously, we all
know that 4 plus 3 is 7.
We're not saying what are we
looking at this problem for.
What we're trying to show now
is a very important aspect
of the game of mathematics.
You see, notice that in
our list of definitions,
no place do we have the
sum 4 plus 3 defined.
What we are interested
in, now, is not
so much the truthful
statement that 4 plus 3 is 7,
but whether the result
4 plus 3 equals 7
follows inescapably from the
definitions that we've listed.
Now, you see, the point is all
we've listed are definitions.
We haven't told any particular
rules of the game yet.
Well, let's make up some rules
as we go along, and as I say,
we'll discuss these in
more details in our notes
and in the exercises.
We essentially do
something like this.
We write down 4 plus
3, then we say, OK, we
do know how to add
by ones; that's
what our definition says.
So we rewrite 3 as 2 plus 1.
In other words, we substitute
2 plus 1, which is equal to 3
by definition.
Then we say, OK, 2 plus 1
is the same as 1 plus 2.
Now, by the way, notice
the tacit assumption
that we're making.
We're assuming that the order
in which you add two numbers
makes no difference.
Obviously, this is not a
rule in every game of life.
In most things in life,
order does make a difference.
Consider, for example,
the statements
first I undress, and then I take
a shower; or first I shower,
and then I undress.
Without meaning to pass
judgment as to which is proper,
at least notice there is a
difference between the two.
What we're saying is
that, somehow or other
in the game of
arithmetic, we assume
that addition has the property
that the order in which you add
makes no difference.
So we say, OK, let's accept
that as a rule of the game.
If we accept that as a
rule the game, 2 plus 1
could then be substituted
for by 1 plus 2.
We make the
additional assumption
that, when you
add three numbers,
the answer does not depend
on voice inflection.
In other words that
4 plus 1 plus 2
is equal to 4 plus 1 plus 2.
And the strategy
behind doing that
is that we know that another
name for 4 plus 1 is 5.
In other words, we now arrive
at the fact that 4 plus 3
is equal to 5 plus 2.
We now rewrite 2 as
1 plus 1, and we now
have the 5 plus 2 is the
same as 5 plus 1 plus 1.
We now again use the fact
that voice inflection
makes no difference,
and we rewrite this as 5
plus 1 plus 1.
5 plus 1, we know by
definition, is 6, and 6 plus 1,
we know by definition, is seven.
In other words, subject to the
rules that we've talked about
implicitly here but have not
stated in our game format
explicitly, what
we have shown is
that if we accept certain rules,
it follows from our definitions
that 4 plus 3 equals 7 is
an inescapable conclusion.
Notice that the inescapability
of the conclusion
hinges on the fact that
we've accepted certain rules.
If we change the rules, we
can change the conclusions.
In other words,
this thing called
drawing inescapable conclusions
is something called validity,
and validity involves the art of
drawing inescapable conclusions
using given rules and
given definitions.
We'll talk about
that in more detail,
but for now, notice
the difference here.
Before we did this, we knew as
a conjecture, a past experience
thing, that 4 plus 3 equals
7 is a true statement.
What we now know is in terms
of certain rules of the game,
coupled with the definitions
that we've accepted, 4 plus 3
equals 7 is an inescapable
conclusion, henceforth to be
called a "theorem" in our game.
Let's see if we can't get
away from this rather simple
example, and again, let me
emphasize that as simple
as this example is,
throughout our course,
we will be using the
same technique, only
at a more sophisticated
level of computational skill.
But let's take a look here and
see what we're really saying.
What mathematical
structure really involves
is a logic machine
type of thing.
We have a logic
machine, a machine
that, being fed any kinds
of definitions, rules,
assumptions, et cetera, grinds
these things through and has,
as its output,
inescapable conclusions.
In other words, you feed in
definitions, rules, et cetera,
which by the way,
don't really have
to be true in the real-life
sense of being true.
I mean, for example,
in the baseball game,
to say the rule is
three strikes is an out,
there was certainly
no basic truth
the said that had
to be the case.
What is true is that,
in the world of science,
the scientist is the
interpreter of nature.
What happens in real life
happens whether the scientist
predicts it or not.
What he tries to do is to make
up definitions and rules, which
are compatible with
his experience, things
which he calls truth.
He feeds those through
his logic machine,
draws inescapable conclusions,
and if the conclusions
follow inescapably from the
definitions and the rules,
we call the resulting
argument valid.
In other words, truth
is a value judgment
that we make about a
particular statement.
Validity is a more
objective thing
that we attribute
to an argument.
In other words, a statement
is either true or false.
An argument is either
valid or invalid,
meaning that we only judge
whether the conclusion
of the argument follows
inescapably from the given
assumptions, independently
of whether those assumptions
happen to be true or not.
By the way, as an aside,
one of the reasons
that the scientist
prefers to dodge issues
such as "what is truth" and
leaves that to the philosopher,
is that truth, in many
cases, is a relative thing.
It's based on the
available knowledge.
It's also based on the situation
that we want to handle.
In other words, what is truth?
And my claim is that the answer
depends on the situation.
Well, let me give you again a
trivial arithmetic situation.
Does 1/2 plus 1/3 equal 5/6,
or does 1/2 plus 1/3 equal 2/5?
And again, the
answer is it depends
on what real-life problem
you're dealing with.
For example, if a person is
working for you by the hour,
and he works for you
for a half hour one day
and a third of an hour the next
day, the total time he's worked
is 5/6 of an hour.
Why?
Because this agrees with
our real-life experience
that 30 minutes plus 20
minutes is 50 minutes.
On the other hand,
if a baseball player
goes one for two in the
first game of a doubleheader
and one for three in the
second game of a doubleheader,
he has batted two hits
in five times at bat.
In fact, if you could
convince the public
that he had five hits
and six times at bat,
you could become a director
of any economy program
in the nation, I guess.
The point, however, is this.
In most arithmetic
questions that we deal with,
we are used to the
physical interpretation
in which 1/2 plus 1/3 equals
5/6 reflects the real life
situation.
Whereas this example here
may seem trivial, namely,
how often are you going to be
involved with batting averages
unless you're doing
sixth grade arithmetic.
The point remains, ironically
or whatever you want to call it,
that this little batting
average problem is not trivial.
In the world of engineering,
we know this example
as the weighted average problem.
In other words, you'll notice
that 2 over 5, as a fraction,
is more nearly equal to 1/3
that it is equal to 1/2,
and the reason for that
is that the player batted
at the low average, 1 over
3, one hit in three times
at bat, for more times at bat.
In other words, the three has
a heavier weighting factor
than the two, and every time
that you use a weighted average
in a scientific engineering
oriented investigation,
this is the truth, not this.
In other words, it is neither
true or false that 1/2 plus 1/3
equals 2/5 or that 1/2
plus 1/3 equals 5/6.
Which is true depends on the
particular physical situation.
You see a rather
interesting point here,
that we sometimes
allow truth to be
based on what particular
problem we're trying to solve.
Another way that
we manipulate truth
is that we sometimes have a
rule that we like to be true.
In fact, I guess
this is probably
what happens with most
political theories
that one starts
with the objectives,
knows what it is that
he wants to be true,
and then invents the
definitions and the rules
to conform with this.
In much the same way as
around the fourth grade again,
we learn such adages as
"Look before you leap,"
and two minutes later you learn
"He who hesitates is lost,"
and suddenly you come
to the conclusion
that you can make your
assumptions validify
any conclusion you want just
by choosing your assumptions
appropriately.
Now, if that sounds
degrading, let
me show you how it's used
effectively in mathematics.
In other words, let me just say
that predetermined rules may
control truth.
Let me give you an example.
Going back to exponents,
why does b to the 0 equal 1?
The answer is very
simple, that when
we use positive exponents,
positive whole number
exponents, we talked about
that many factors of b.
And one of the
interesting rules that we
saw that was obeyed by
whole number exponents
was that if you multiply
b to the m-th power
by b to the n-th power,
you got as an answer b
to the m plus n power.
b to the m plus n.
Now, the interesting thing
is that after a while,
you never even paid attention
to why this rule worked.
What you did know was that
this was a pretty darn
convenient rule to use.
Computationally,
this rule simplified
many particular computations
that you were doing.
In particular, then,
as soon as n is 0,
you would still like to
be able to use this rule.
In other words, we
want this nice rule
to apply even when n equals 0.
Now, let's look at this from
a computational point of view.
If we want this rule to
apply when n equals 0,
let's simply
rewrite this exactly
as it appears here
with n equal to 0.
We then have, what?
We have b-- we'll just
repeat everything here.
b to the m times b.
Now, we're replacing n by
0, so that's b to the 0,
and that must equal b to the
m plus n, that's m plus 0.
But the interesting
point is that we
know how to add numbers, and
for numbers, m plus 0 is m.
In other words, this
is still b to the m.
Now, we look at this,
and what we're saying
is if we want the rule b
to the m times b to the n
to equal b to the m plus n
to be true even when n is 0,
this says that b to the 0
must be that number such
that when we multiply it by b
to the m, we get b to the m.
Now, what number
has the property
that when you multiply it by
b to the m you get b to the m?
And if you're real
quick, you say 1,
and if you're
algebra-oriented, you
say b to the 0 is therefore
equal b to the m divided
by b to the m, and
again you say 1,
except of course that b must
be unequal to zero because,
hopefully, by this time we
understand why we must never
divide by 0.
In other words, what we now
have is the old high school
rule that b to the 0 equals
1 provided b is unequal to 0.
But here's the important point.
If I have never
defined b to the 0,
and somebody says
to me: "make up
a definition for b to the
0," and I say, OK, b to the 0
is going to be 36, I have
every right in the world
to make up that definition.
What I don't have
a right to do is,
when I'm ever using an
expression like b to the 0,
is to assume that
I have the right
to use this particular recipe.
In other words, if I want to
be able to use the nice recipe,
even when n is 0, I have no
choice but to define b to the 0
to equal 1.
Therefore, we make
up that definition
because we want that
recipe to apply,
and that's exactly what
we're going to be doing
through most of this course.
We are going to look at
certain real-life situations,
we are going to look at certain
recipes that we want to apply,
and we are going to make
up definitions this way,
and then see how our
inescapable conclusions follow
from these definitions.
More about that will be said
in the remainder of this unit,
and our next
lecture will pick up
the game of mathematics
in a different context.
But until next time, good bye.
Funding for the
publication of this video
was provided by the Gabriella
and Paul Rosenbaum Foundation.
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