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.
We do it all the time now.
When you have any sort of task,
you’ll find that you’re actually using some sort of perspective.
Suppose that you’re hiring someone.
And you’ve got a bunch of recent college graduates who apply for a job.
And you’ve gotta think,
“Okay, how do I organize all these applicants?”
Let’s say 500 applicants.
One thing you could do is you could organize them by GPA:
Take the highest GPA down to the lowest GPA.
That’s be one representation.
And you might do that if you valued competence or achievement.
But you might also value work ethic.
And if that were the case you might instead organize
those same CV’s or application files by how thick they are.
[Those who’re going to do the] really thick ones are people who work really, really hard.
They’ve accomplished a lot.
Well, the third thing you might do is you might value creativity.
And you might say,
“Well, let’s put the ones that are sort of most colorful, most interesting over here.
And the ones that are least colorful and least interesting over here.”
That’s the third way to do it.
Now depending on what you’re hiring for,
depending on who the applicants are,
any one of these might be fine.
The only point I’m trying to make here is that there’s different ways to organize these applicants.
In each one of those ways you organize—
whether it’s in your head,
or whether it’s formally laying them out in some way—
is a perspective.
And those perspectives will determine how hard the problem will be for you.
Let me explain why.
Now I want to go back to the landscape metaphor.
And when I think of that landscape as being rugged,
and by rugged I mean that it doesn’t look like a single peak,
that there’s lots of peaks on it.
And I want to formalize this notion of peaks.
And I do so as follows:
I’m going to define what I call a local optima.
A local optima is a point such that
if you look at the points on either side of it,
they’re lower in value.
So it’s sort of a point that locally is the highest possible value.
So if I look at this particular rugged landscape again,
there’s three local optima: 1, 2, 3.
At any one of these three points, I’d be stuck:
If I looked to the left or to the right,
I wouldn’t find a solution that’s better.
So we think about what makes a good perspective:
A good perspective is going to be a perspective that doesn’t have many local optima.
A bad perspective is going to be one that has a lot of local optima.
Let me give you an example, okay?
So, suppose I’m coming up with a candy bar.
Suppose I’m tasked with coming up with a new candy bar.
So I have my team of chefs make a whole bunch of different confections for me to try,
and I want to find the very best one.
But there’re so many of them, there’s so many possibilities,
that I’m not even sure how to think about it.
But one way to represent those candy bars might be by the number of calories that they had.
So I can organize all the different things they make by number of calories.
And if I did that, maybe I’d have three local optima.
So that’s a reasonable way to represent these possible candy bars.
Alternatively, I might represent those candy bars
by masticity, which is chew time—
how long it takes to chew ’em.
So these would be the ones that maybe only take two minutes to chew.
And these may take twenty minutes to chew.
Well, chew time is probably not the best way to look at a candy bar.
And so, as a result, I’m going to have a landscape with many, many more peaks.
And so, because it’s got many more peaks, that’s more places I could get stuck.
So it’s not as good as a way to represent the possible solutions.
It’s not as good a perspective.
The best perspective would be what we call a Mount Fuji landscape,
the ideal landscape that just has one peak.
And these are called Mount Fuji landscapes
because if you’ve ever been to Japan,
and you look at Mount Fuji, it looks pretty much like this.
Actually not quite like this, there’s like snow on the top.
But for the most part, it looks just like one giant cone.
If you’re on a Mount Fuji landscape,
if you’re sitting at some point,
you can always just climb your way right up to the top.
So these single-peak landscapes are really good
because you’ve basically taken a problem
and made it very, very simple.
What would be an example of a Mount Fuji landscape?
I’m going to take a famous example.
So, a famous example comes from scientific management,
and due to Frederick Taylor.
Taylor famously solved for the optimal size of a shovel.
So let’s think about the shovel size landscape.
So, on this axis, I’ve got the size of the shovel.
And on this axis, I’ve got the value.
And what do I mean by the value?
I don’t mean how much I can sell the shovel for,
I mean it’s like how useful the shovel is at the task.
So let’s suppose we’re shoveling coal
and I want to think about
how many pounds of coal can some[one] shovel in a day
as a function of the size.
So let’s start out here where the size is zero.
So this is the size of the pan.
If I have a shovel has a pan of size zero,
that’s commonly known as a stick
and we can’t get anything.
We’re not going to shovel anything with a stick.
Well, if I make it bigger,
you know, make it the size of maybe like a little spoon or something,
then we can shovel a little bit.
And as I make the shovel bigger and bigger and bigger,
we, whoever, my workers, can shovel more and more coal.
But at some point, the shovel’s going to get a little bit too big.
And it’s going to be too heavy to lift.
And the worker’s going to get tired,
and I’ll shovel less,
he’ll shovel less and less and less and less.
And then eventually get to some point where the shovel’s so big
that he can’t even lift it,
and it’s as useless as the stick.
So if I look at value in terms of how much coal the person can shovel in a day is a function of the size of the shovel.
I’m going to get a single-peaked landscape.
That’s going to be an easy problem to solve.
And this idea, that we could represent scientific problems in this way—
or we could put engineering problems in this way—and then climb our way to peaks,
is the basis is something called scientific management
And the idea was that you could then
by finding these high points on these landscapes,
find optimal solutions.
We’re only going to find out the optimal solution for sure
if your hill climbed like this—if it’s single peaked.
If it’s rugged and looks like this mess,
looks like Mount Fuji landscape you’re fine,
but if it looks like this mess, this masticity landscape,
if you have a bad perspective,
well then if you climbed hills
you could get stuck just about anywhere.
So what you’d like is you’d like a Mount Fuji landscape,
And in the case of simple things like this shovel, that’s easy to get.
Let me give you another example.
This one’s a lot of fun.
This is a favorite game of mine called Sum to fifteen
and was developed by Herb Simon
who’s a Nobel Prize winner in economics.
And Sum to fifteen was developed to show people
why diverse perspectives are so useful,
why different ways of representing a problem can make them easy,
can make them like Mount Fuji,
or can make them really difficult.
So here’s how Sum to fifteen works.
There’s cards numbered from one to nine face up on a table.
There’s nine cards in front of you.
There’s two players.
Each person.takes turns, taking a card.
until all the cards are gone, possibly—it could end sooner.
If anybody ever holds three cards that add up to exactly 15, they win.
That’s the game. So, really simple.
Nine cards. Alternate taking cards.
If you ever get exactly three that sum to fifteen you win.
So let me show you a game.
Here’s a game between two people,
[let’s] call them Paul and David.
Paul goes first. Now you’d think when you play this game
the thing to do would be to choose the five.
Paul chooses the four, which is sort of an odd choice.
David goes next so he takes the five.
Paul then takes the six.
Now the six is a strange choice
because four plus six plus five equals fifteen.
So it looks like there is no way that he can win.
Well this will be confusing to Doug.
So Doug’s going to take the eight.
Now notice eight plus five equals thirteen.
So that means Paul has to take the two.
So he takes the two.
Well think about what happens next:
Four plus two is six.
So if Doug doesn’t take the nine, he’s going to lose.
But six plus two is eight.
So if Doug doesn’t take the seven he’s going to lose.
So what you’ve got here is that Paul has won.
No matter what Doug does, Paul’s going to win the game.
Now this is a pretty tricky game, right?
It was developed by a Nobel Prize winner.
You could imagine there’s lots of strategy involved.
I want to show you this game in a different perspective.
Remember the magic square from seventh grade math?
Every row adds up to fifteen—
8+3+4, 1+5+9, 6+7+2 —
so does every column—
8+1+6 sums up to fifteen;
3+5+7 sums up to fifteen—
and even the diagonals—
eight, five, two is fifteen;
six, five, four is fifteen.
Every row, every column, every diagonal sum up to fifteen.
Let me show you this game again on the Magic Square.
So, it’s just a different perspective on “Sum to Fifteen”.
Paul goes first, and takes the four.
Doug goes next and takes the five.
Paul takes the six, which is an odd choice, because now he can’t win.
Doug then takes the eight, Paul blocks him with the two.
But now it turns out, either the nine or seven will let Paul win.
What game is this?
Well, you’re right, it’s tic-tac-toe.
Sum to fifteen is just tic-tac-toe,
but on a different perspective, using a different perspective.
So if you turn Sum to Fifteen—
if you moved the cards 1 to 9 and put them in the magic square—
what you do is you create a Mount Fuji landscape In a sense:
You make the problem really simple.
So a lot of great breakthroughs,
like the periodic table,
Newton’s Theory of Gravity,
those are perspectives on problems
that turned something that was really difficult to figure out
into something that suddenly makes a lot of sense,
very easy to see the solution.
At least it’s something I call in my book, one of my books, the difference,
I call this the Savant Existence Theorem.
For any problem that’s out there,
there exists some way to represent it,
so that you turn it into a Mt. Fuji problem.
Now, why is that?
Well, it’s actually fairly straightforward.
All you have to do is,
if you’ve got all the solutions here represented on this thing,
you put the very best one in the middle.
And then put the worst ones at the end.
And then just sort of line up the solutions in such a way
so that you turn it into a Mount Fuji.
So it’s very straightforward.
Now the thing is, in order to make the Mount Fuji,
you’d have to know the solution already.
This isn’t a good way to solve problems
but the point is, it exists.
So it’s always the possibility
that someone could look at particular problem and said,
“Hey, what if think of it this way?”
And doing so turn something that was really rugged
into something that looks like Mount Fuji.
Here is the flip side though.
There is a ton of bad perspectives.
So just like there’s these Mount Fuji perspectives,
there’s also lots and lots of horrible ways to look at problems.
Think about this: Suppose I have just ten alternatives
and I want to think about what are all the different ways I can just put them in a line.
Well there’s ten things I could put first,
nine things I could put second,
eight things I could put third and so on.
So there’s 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 perspectives.
Most of those are going to not be very good.
They’re not going to organize this set of solutions in any useful way.
Particularly, only a few of them are going to create Mount Fujis.
So we think about the value of perspectives, what we get is this:
There’s really good ones out there,
that insightful, smart people can come up
with really good representations of problem[s]
to make the landscapes less rugged.
If we just think about things in random ways,
we’re likely to get a landscape that’s so rugged
that we’re going to get stuck just about everywhere.
We’re not going to be able to find good solutions to the problem.
And we’re going to hit things that look like the masticity landscape,
and we’re going to get things with lots and lots of peaks.
Let’s move on now and talk about how we move on these landscapes.
So once I got our landscape, how do I find better solutions?
Are there other alternatives to just sort of climbing a hill?
Because that hill climbing idea really only works in one dimension.
What if I’ve got all sorts of dimensions?
How do I think about…
(Just a sec…)
So what have we learned?
First thing we’ve learned is that when we go about trying to solve a problem,
when we encode it in some way,
that’s a perspective.
And a perspective creates peaks; it creates these local optima.
So a better perspectives have fewer local optima.
Worse perspectives have lots of local optima.
And if you think about how many perspectives are out there,
we just saw there’s billions of them.
Because there’s billions of perspectives,
most of those probably aren’t very useful.
Some of them, though, turn problems into Mount Fujis.
And sometimes it takes a genius—
it takes a Newton, it takes a Mendeleev—
to come up with a way of representing reality
so that something that was incredibly rugged
becomes Mount Fuji–like.
Other times, if you think about the size of a shovel,
that problem most of us could probably figure out a way that problem just by shovel size,
so that it becomes a Mount Fuji.
The big point is this:
When we go about solving problems, the first thing we do is we encode them.
We have some representation of the problem.
That representation determines how hard the problem will be.
If we represent it in such a way that it’s a Mount Fuji, it’s easy.
If we represent it in such a way that it looks like that masticity landscape,
it’s probably going to be fairly hard.
Where we want to go next,
is we want to talk about once we’ve got this representation of the possible solutions,
once we have that landscape, so to speak,
how do we search on that landscape?
So one thing we’ve talked about was climbing hills.
But there’s lots of different ways you can climb hills.
That’s what we’ll talk about next: the heuristics we use on a landscape.
Thanks.