Hi. In this lecture we’re talking about problem solving.

And we’re talking about the role that diverse perspectives play in finding solutions to problems.

So when you think about a problem,

perspective is how you represent it.

So remember from the previous lecture, we talked about landscapes.

We talked about landscape being a way to represent

the solutions along this axis

and the value of the solutions as the height.

And so this is metaphorically a way to represent

how someone might think about solving a problem:

Finding high points on their landscape.

What we want to do is take this metaphor and formalize it

and part of the reason for this course is to get better logic,

[in order to] think through things in a clear way.

So I’m going to take this landscape metaphor and turn it into a formal model.

So how do we do it?

The first thing we do is we formally define what a perspective is.

So we speak math to metaphor.

So what a perspective is going to be is

it’s going to be a representation of all possible solutions.

So it’s some encoding of the set of possible solutions to the problem.

Once we have that encoding of the set of possible solutions,

then we can create our landscape by just assigning a value to each one of those solutions.

And that will give us a landscape picture like you saw before.

Now most of us are familiar with perspectives,

even though we don’t know it.

Let me give some examples.

Remember when we took seventh grade math?

We learned about how to represent a point, how to plot points.

And we typically learned two ways to do it.

The first way was Cartesian coordinates.

So given a point, we would represent it

by and an X and a Y value in space.

So, it might be five units,

this would be the point, let’s say (5, 2).

It’s five units in the X direction, two units in the Y direction.

But we also learned another way to represent points,

and that was [polar] coordinates.

So we can take the same point and say,

there’s a radius, which is its distance from the origin,

and then there’s some angle theta,

which says how far we have to sweep out,

in order to sweep that radius out in order to get to the point.

So two totally reasonable ways to represent a point:

X and Y, R and theta.

Cartesian, polar.

Which is better?

Well, the answer? It depends.

Let me show you why.

Suppose I wanted to describe this line.

In order to describe this line I should use Cartesian coordinates,

’cause I can just say Y=3 and X moves from two to five.

It’s really easy.

But suppose I wanna describe this arc.

If I wanna describe this arc,

now Cartesian coordinates are gonna be fairly complicated,

and I’d be better off using polar coordinates,

because the radius is fixed

and I just talked about how the radius is—you know,

there’s this distance R,

and theta just moves from, you know, A to B, let’s say.

So depending on what I want to do.

If I want to look at straight lines, I should use Cartesian.

And if I want to look at arcs, I should probably use polar.

So, perspectives depend on the problem.

Now let’s think about where we want to go.

We want to talk about how perspectives help us find solutions to problems

and how perspectives help us be innovative.

Well, if you look at the history of science a lot of great breakthroughs—

you know, we think about Newton,

you know, his theory of gravity—

you can think about people actually having new perspectives on old problems.

Let’s take an example.

So, Mendeleev came up with the periodic table,

and in the periodic table he represents the elements by atomic weight.

He’s got them in these different columns.

In doing so, by organizing the elements by atomic weight

he found all sorts of structure.

So all the metals line one column, stuff like that.

Remember—from high school chemistry class.

That’s a perspective: It’s a representation of a set of possible elements.

He could’ve organized them alphabetically.

But that wouldn’t have made much sense.

So alphabetic representation wouldn’t give us any structure.

Atomic weight representation gives us a lot of structure.

In fact, when Mendeleev wrote down

all the elements that were around at the time according to atomic weight,

there were gaps in his representation.

There were holes for elements that were missing.

Those elements became scandium, gallium, and germanium.

They were eventually found ten to fifteen years later,

after he’d written down the periodic table:

People went out and were able to find the missing elements.

That perspective, atomic weight,

ended up being a very useful way to organize our thinking about the elements.