Productive Thinking

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roductive Thinking...the Gestalt Emphasis

Productive Thinking...the Gestalt Emphasis

Lecture Material

q Illusions and Ambiguous Figures

q Organizing Principles of Perceptual Grouping

r Introductionr Wertheimer Perceptual Grouping

q Partitioning the Physical World into Objects

r Some Natural Scenes

r Some Artificial Scences

r A Procedure that Utilizes Physical Constraints

q Problem Solving Representation and Constraints

r Cryptarithmetic Problem

r Checkerboard Problem

r Mutilated Checkerboard Problem

r MatchMaker Problem

Terms, Concepts, and Questions

q Terms, Concepts, and Questions

Assignments/Exercises

q Traffic Lights

q !Cryptarithmetic Problem

q Matchstick Problems

Table of Contents

Charles F. Schmidt

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lusions and Ambiguous Figures

Does the mind always represent the world accurately and unambiguously?

rceptual illusions and ambiguous figures were of special interest to the Gestaltists. Artists have also been fascinated

ese perceptual phenomenon. Perceptual illusions and ambiguous figures are of special interest in the investigation of

nking because:

q illusions seem to indicate that our mind does not always accurately represent the perceptual input. For the

Gestaltist, this suggested that the mind was "actively" involved in interpreting the perceptual input rather than

passively recording the input.

q ambiguous figures exemplify the fact that sometimes the same perceptual input can lead to very different

representations. Again, the Gestaltist took this as suggesting that the mind was actively involved in interpreting

input.

q what I will call completion figures are figures which the mind rather unambiguously interprets in a particular w

despite the fact that the input is incomplete relative to what is typically "seen"

hat follows are a variety of examples of these phenomena which you can select and explore.

lusions

q Mller-Lyer Illusion

q Hering-Helmholtz Illusion

q Ebbinghaus Illusion

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lusions and Ambiguous Figures

Ambiguous Figures

q Rubin Vase

q Old/Young Woman

q What is this a photo of?

ompletion Figures

q Triangle Completion

ome Artists' Versions (click to enlarge)

Escher Riley Magritte


Charles F. Schmidt

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Untersuchen zur Lehre von der Gestalt - 1

Wertheimer on organizing principles of perceptual grouping

Max Wertheimer was one the more famous gestalt researchers. Gestalt can be translated as "form" and part of the

mphasis of the gestalt group was that "the whole (or gestalt) was greater than the sum of its parts." We aren't interes

re in exactly what this slogan was meant to convey. However, notice that this represented an explicit focus on the

estion of composition; that is, how are our ideas structured. The slogan was meant to convey the gestaltist's belief s process of composition probably wasn't a simple one.

his slogan can be understood a bit better if we get a little more technical. Summation is, of course, an additive proce

you recall the implications of the axioms of addition, you may recall that the contribution of one number to the sum

dependent of when you add it...or put another way, where it appears in the summation. E.g. 2 + 3 + 4 = 9 = 3 + 2 +

hat is, the 3 that appears in bold, and in fact the whole equation yields the same result regardless of the context. And

ct 3 is still three in the equation 1,899, 622, + 3 + 567,333. The gestaltist were suggesting that we really had to con

e context in which some element occurred in order to understand how it contributed to the whole or gestalt. They

ought that the mind rarely combined things or organized things independent of the context in which they appear.

onsider the two configurations shown below:

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Untersuchen zur Lehre von der Gestalt - 1

In both of these configurations we have a larger rectangle placed between two smaller rectangles. But note that th

arge rectangle" in the top figure is exactly the same size as the two "smaller rectangles" in the lower figure. Here th

ontext" is presumably influencing the way in which I encode the various rectangles.

The fact that there are two ways of viewing configurations of this sort was very important to the arguments that w

between the gestaltists and behaviorists during the earlier part of the 20th century. On one view, an absolute or co

dependent view, we place the center rectangle of the top figure and the left and right rectangles of the lower figure

e same equivalence class because they are exactly the same size and shape. But, on another view, the middle rectan

each figure should be seen as equivalent because they both depict a large rectangle that is between two smallerctangles.

The fact that there are two possibilities may pose a problem for a "passive mind". The environment can hardly be

ked to decide which of these alternatives is seen by the viewer. The "stimulus" seems to be a somewhat ambiguous

tion when thought of in this way...But, since the behaviorist studied the way in which stimuli became associated w

sponses, the behaviorist couldn't allow the stimulus to be a fuzzy notion. (Note that ambiguous figures present exac

e same problem and this is one of the reasons that they have been studied so extensively in perception.)

Recall that I referred to the large rectangle between two smaller rectangles. Relations such as 'between' also gave

some difficulties. Is "between" something that is out there in the world? Can you point at it? Well, not really. It seebe a way in which we describe a situation and not an intrinsic property of the situation. (Cf. the Wrzburg Group

scussed in Chapter 1 of your text.)

And if that isn't bad enough, I can also correctly say things such as: "There is not a circle in either picture." "Ther

t a triangle in either picture." There is not an apple in either picture." etc. ad nauseum.... Even this simple innocuou

ery day word "not" causes problems since we can use it to make true statements about the world. But, how does th

uation or stimulus elicit statements of this sort....there are infinite number of such statements that I could make abo

y situation!

Wertheimer on Grouping of Elements Productive Thinking...the Gestalt Emphasis

Charles F. Schmidt

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Untersuchen zur Lehre von der Gestalt 2

On Perceptual Grouping

Adapted from Max Wertheimer, Untersuchen zur Lehre von der Gestalt, II. Psychologishe Forshung, 1923, 4,301-305.

Now we turn to Wertheimer's consideration of perceptual grouping. What is most important about this work in th

present context is that Wertheimer attempted to see what our mind does against a consideration ofwhat our mind

might have done. In the work we review here, Wertheimer explores the way we typically perceptually group elemen

But in order to determine whether the mind is exerting a "bias" he considers some of the ways these same elements

could have been grouped but weren't. The behaviorists typically weren't this analytic. Consequently, it was the gestal

hat kept trying to point out these pesky problems that can arise when you are a bit more analytic about what you are

o.

The examples that Wertheimer constructed were very simple; most of them consisted of a set of dots. The purpose

of these examples was to aid in the understanding of the different factors that influenced the grouping or composition

elements into wholes. The gestaltist had suggested what they called unit forming factors that influence how elements

are grouped or organized in wholes. These unit forming factors were:

q similarity;

q proximity;

q common fate;

q good continuation;

q set; and

q past experience.

The examples that follow are mainly concerned with the first two factors. The first example shown below consist

a row of 10 dots. Think of labeling these dots from left to right as a, b,...,j. Below this row of dots is shown two ways

which the dots might be organized. The one on the left is in terms of groups of two as indicated by ab/cd/.... where th

elements that are close to each other are grouped into a unit. An alternative basis for grouping, depicted on the right b

a/bc/de/... involves grouping the leftmost and rightmost elements of adjacent pairs together; e.g. bc. This is a logical

possibility; and, if you work at it you may be able to "see" that organization. But, it is not the organization that we

would typically see; nonetheless, we can imagine a mind that would be biased to see this latter organization of the do

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We have shown only two ways in which the dots might be organized. In fact, there are many possible ways. Ther

are 10 dots in this example. If we think of these 10 dots as being members of a set, then what we have called a group

s simply a partition of this set. The number of possible partitions of a set ofn items is given by something called a

Stirling number, and these numbers can get large very rapidly. To get some idea of why this is the case, consider

omething called the power set. The power set is simply the set of all possible subsets of a set. For example, the pow

et of the the set {x,y,z} is the set { {x,y,z}, {x,y}, {x,z}, {y,z}, {x}, {y}, {z}, {} }. Note that a set is a subset of itsel

and every set has the empty set, {}, as a subset. Now, a partition of a set is simply a selection of a set of subsets such

hat no element occurs in more than one of the subsets selected and each element occurs in one of the selected subset

For example [ {x,z} ,{y} ] is a partition. Note that there are 8 sets in the power set of this set of three elements. Ingeneral, a set of size n has 2 to the n subsets. The set of 10 dots below has 2 to the 10 subsets, and then there are all th

ways to choose from these 1024 subsets to form partitions! In case your interested in seeing a list of the power set of

he 10 element set {a,b,c,d,e,f,g,h,i,j}, click here.

The next example below is similar except that now the dots are arranged in a diagonal. Again, two of the many

possible groupings are shown with the one that we are biased to use appearing on the lower left.

The figure below has increased the number of dots to 15, but, despite the enormous increase in the number of

possible groupings, we don't see this as very different from the 10 dot figure above



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In order to help you visualize these groupings, the two groups above have been animated as well as a random

grouping. To see each of these animations, click on the selections below.

In the next example below, it is clear that the distance between the dots influences whether we organize it as rows

columns.

In the top matrix, all of the dots are equidistant and you can probably organize them in a number of ways. But the

matrices below show that the similarity in "color" is a another factor that influences the way we group these element



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In the examples below, we tend to group the dots as forming two lines; e.g. in the left and middle figures we tend

ee the A and C segments grouped as a single "line" and the B segment grouped as a line; rather than the A, B, and C

egments each as a line.

The next two figures, one composed of dots and the other of continuous lines further demonstrate our bias in

grouping such figures.



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The next example illustrates the factor that the gestaltists referred to as common fate. Here the arrows pointing in

common direction tend to be grouped.

Finally, the examples below provide evidence for the effect of past experience and context on the grouping of

elements. Notice that in the top figure the middle lines are grouped into single whole, but in the next figure they are

grouped into two elements.



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And, finally the image below shows only the center element which is, of course, the same in both examples.


Charles F. Schmidt



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artitioning the Physical World

REALITY AND POSSIBILITY:

Seeing Reality by Imagining the Possibilities

"In order to determine what happened, imagine all of the

possibilities and when you have eliminated all but one, then no

matter how improbable, that remaining possibility is the truth."

(Sherlock Holmes' Father, I think.)

Think of clouds in the sky, waves in the sea, and flags in the wind. Describe their shape. The pictures below provide a

napshot of the shape of clouds, waves and flags as caught by the camera for a brief moment.

Recall that in the previous section we considered the

question of how we organized simple figures such as sets dots. When there were ten dots we noticed that we groupe

them into an organization that was referred to as a partitio

A partition is a special way in which to divide a set into

subsets. A partition is a set of subsets of a set such that eac

member of the set is in one and only one subset. And, each

member of the set is in some subset. With our previous

examples of dots and lines this seemed like such an obviou

condition to impose that you probably didn't even think ab

it.

But study the clouds in the picture above of the coast nCan Cun and the sail boats and breaking waves near Diam

Head on Oahu in the picture below. How many clouds are

there? How many waves?

Notice that in order to try to answer the question of 'ho

many' you need to establish exactly where the borders are

separate one cloud from another and one wave from anoth

Not an easy task even in a snapshot; and the reality is muc

more complicated since had the snapshots been taken a few

instants later, an entirely different configuration of clouds

waves would be recorded.

Clouds and waves have a shape and at times can be

individuated; but the individuals don't persist. The shape

changes and the identity of the momentarily individuated

soon disappears forever in the constant flux of change. Bu

what about the flags shown in the next picture below?

In this case it is quite easy to individuate these flags fly

over Fort Sumner. And their shape? Clearly they are

rectangular in shape.

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But is this so obvious? Or is this an inference not draw

from the visual information alone but also drawn using ou

information about how flags are made?

The flag poles also help with this inference. If the last f

pole at the far left were not visible would you still be as

certain that there are exactly six flags in the picture? Is it

really so easy to decide the contours to those flags on the f

left? Or, put in a way that harks back to the idea of a partit

it it really a simple matter to determine for each contour th

one and only one flag that it belongs to?

Notice that. as with the waves and the clouds, the shape

the flags is varying constantly with the vagaries of the win

Is our belief that the flags really have a constant rectangul

shape justified? If so, why wouldn't we be justified in

believing that clouds were actually square and waves

rectangular?

Consider next the picture shown below on the left of onthe newer buildings in Manhattan. What is its shape? How

many buildings are in the picture?

Now look at the picture shown below on the right. Is thi

the same building? What is its shape? How many building

are shown in this picture?

Notice that in this case we assume that the shape of the

building is not changing in the wind and its individuation

secure (Think of the problems establishing property rights

this weren't the case!). What has changed in the two picturthe point from which the building is viewed.



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Additionally, in both pictures various degrees of shadowing create "contours" where we are quite certain there are no

ontours......only shadows. But how do you know this? Have you made some assumptions which have licensed this

onclusion? Also of particular interest in the picture on the right is the trihedral vertex...the point where three lines inters

what is almost the center of the picture. What is going on here? Trihedral vertices will be examined in detail in the nex

age and perhaps we will then understand why this aspect of the picture may seem problematic.

In this section we will examine the role assumptions play in our ability to quickly and quite accurately organize the

omplex visual information that informs us about our everyday world. The particular assumption that is our focus is the

sumption that much of our world is populated with objects that don't change shape despite the fact that their shape appe

change as we move around relative to the objects or the objects move relative to ourselves. This assumption, that many

e objects of our world do not change shape is called "the rigid body assumption". On the next page are various drawin

at will sharpen our visual intuitions regarding partitioning the world of objects.

Next Page on Partitioning the Physical World into Objects Productive Thinking...the Gestalt Emphasis

Charles F. Schmidt



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artitioning the Physical World 2

Some Artificial Figures

It appears that our mind probably has to make the rigid body assumption an assumption about the world in order to

ble to organize the contours of objects into a set of objects. To get some feel for what we are talking about, look at the

ctures shown below which were designed to thwart out ability to quickly organize the world into a set of objects.

Try to organize the mass of connecting lines shown above into a single partition of objects. I created this picture from

asic rectangular wire frame that outlines a rectangular solid. Unless you use certain types of computer graphics progra

ou rarely encounter this type of schematic information. When we view a rectangular solid we see only a subset of the

dges of object.

What you should have experienced, if you are at all like me, is that you could organize these wire frames in a variety

fferent ways...but, you probably weren't able to fix on a single coherent partition of the lines into a set of "objects". E

ough the shape of this object is not changing; it is difficult to come to a fixed organization of the object.

The next picture is more of the same, but I made it larger and a little more complex. If you have a large enough scre

en expand your browser window so you can view the entire picture. Now, try and focus on the center and note the wa

which your mind organizes the lines.

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If this didn't challenge your mind enough, then try the next picture which is more of the same, but lots more and andow all in a jumble. The lack of regularity in the composition will probably make it harder for you to structure this pict



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And, if this hasn't sated your visual appetite, then sink into this next one which is constructed solely from a simple

iangle. You should be able to see many ways in which to organize this picture, some of which are triangles, others

quares and other rectangles.



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Now that you've experienced some of the difficulties your mind has with both photographs of the natural world and

ith these artificially created pictures, we are now ready to ask why the mind doesn't run into these kind of difficulties

f the time.

Well, we don't really know the full answer to this question. But, one intriguing line of research that addresses this iss

so fits in with our emphasis on comparing the logical possibilities with those that actually seem to be considered by th

mind. The discussion here is based on work done by David Waltz in the early 70's on the question of how one might go

om two dimensional information about lines and the way they are connected to an organization of these lines and theionnections into a set of objects.

One way to try to get your mind around this problem is to focus your eyes on some messy area in your current

nvironment. My desk is always available to fill this bill. Notice that the objects overlap each other and only parts of th

urfaces are visible. Now think of simply drawing a line on a sheet of paper for each object edge that you can see. These

nes will intersect and form vertices.

This set of lines and the vertices that they created were what Waltz took as the input to his computer program. The

utput was to be an organization of these lines and vertices into a set of objects....what I referred to above as a partition



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is vertex information into a set of objects. The next page will briefly present the work of Waltz on this partition probl

Waltz Research on Partitioning the Physical World into

Objects


Charles F. Schmidt



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he Waltz Research on Paritioning a Simple Visual World into Objects

Partitioning the World into Objects - Using Physical Constraints

The middle drawing below consists of about 80 lines and 73 vertices (points where lines join). The vertices are indica

n the drawing on the right with a blue circle. In the middle drawing no attempt has been made to make it appear three-

imensional. It is a flat, two-dimensional collection of lines that intersect. Nonetheless, you have probably organized thi

umble of lines and vertices in a way that is consistent with the view that a collection of three-dimensional objects are

epicted in this drawing. The drawing on the left includes the lines that might complete the various objects. The groupi

f the lines and vertices of the middle figure into a set of objects results in a partitioning of the set of lines. A partition

rouping of a set of things into subsets such that no element occurs in more than one subset and each element occurs in o

f the subsets. How might this partitioning be accomplished? The set of possible partitionings is enormous-even for this

ivial example.

Completed "Objects" Basic Lines and Vertices Vertices Explicitly Depicted

Of course, we don't exactly know how the human mind accomplishes this task. But, we can study a simplification of t

sk and thereby:

q obtain a clear understanding of the nature of this problem; and

q demonstrate the way in which assumptions about the world can simplify what is otherwise an intractable problem

The work we will discuss is based on research carried out by David Waltz. His work was more extensive than what w

ll present here. We will limit our discussion of his work to scenes where no more that three lines come together at any

rtex and we will ignore shadows and cracks. Waltz actually considered a more general case, but the story is more simp

ld if we limit ourselves to scenes that only involve trihedral vertices.

We take two rectangles as shown in the figure below on the left and join them to obtain the set of lines and vertices

own in the figure on the right. This figure will serve as our simple example object for use in explaining this work.

Two Rectangle Composed

Example Figure



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Example Figure - Vertices Only

Example Figure - Lines Only

The next figure above on the left shows the example figure without the lines and only the vertices. Note that the three

ghter colored vertices involve two lines and the rest involve three...these are trihedral vertices. Notice that even though

ese vertices are not identical, it appears that there are only a few types represented here.

The next figure above on the right, shows the example figure as only the set of lines that connect the vertices. The

rtices themselves are deleted. Now it turns out that for the simple case that we are considering, the lines must be either

oundary lines or interior lines.For a boundary line, the object lies on one side of the line but not on the other. If we th

walking clockwise around the object on its boundary lines, then we can distinguish two types of boundary lines based

e direction that we are moving. For an interior line, the object lies on both sides. The lighter colored lines in this figure

rn out to be interior lines. We can also distinguish two types of interior lines; those that are convex and those that arencave. The middle white line above is an example of a concave line and the remaining interior lines are convex.

Now, pick up some square or rectangular object, move it around and rotate it while focusing on a particular corner. Y

ll notice that the vertex changes and may even disappear as you rotate the object. There is a systematic way (using a

uclidean three-space and moving the observer relative to this space) to exhaustively identify the types of vertices that ca

cur in the world we are considering.

The figure below illustrates all of the possibilities. Notice that there are only 4 types of junctions or vertices. This doe

t mean that they are physically the same. For example, the size of the angle of an L vertex can vary enormously. But, f

rposes of interpreting scenes composed of objects it is important to see these as equivalent.



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Notice that I have labeled this image as depicting the physically possible vertices; and there are only 18 of them! Wh

e the logical possibilities? The L vertex has two lines, and each line could be labeled in one of 4 ways. Thus, there are gically possible L vertices. The other 3 types of vertices each have 3 lines. Again, each could be labeled in one of 4 wa

hus, each of these vertices admits 64 possibilities. Then, the total logical set of possibilities is 16 + (3 x 64) or 208. Thu

building in knowledge about what is physically possible...by making the so-called "rigid body assumption" we can

astically reduce the possible labelings that must be considered.

Now, recall the obvious fact that any line connects two vertices. And, of course, a line can only be labeled in one way

hus, a decision about how to label one particular vertex affects the other vertex which shares the line with that particula

rtex. When making one decision influences another decision, we say that there is a dependency between the decisions

at one decision constrains the other. Sometimes, as in the traffic light example you worked out, the constraint is so stro

at knowing one decision completely determines another. Sometimes it is weaker, and knowing one decision precludes

hers, but does not uniquely determine the other.



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The figure above shows our example object with each vertex identified as to its type. Below the object is a vertex by

rtex table where a cell entry is 1 if the two vertices share a line. I have also referred to this as a communication matrix.

his term is used because in communication if I can communicate something to you, then it may be that you will

mmunicate my message to someone else. If you do, then I have indirectly communicated with that person. Things are

alogous here. For example, referring to the matrix and picture above, notice that L1 shares a line with A1 but not with

ut A1 shares a line with L2. Consequently, a decision about how to label L1 will affect L2 even though there is not a dinnection. The algorithm that works out all of these effects is called a constraint propagation algorithm. The animatio

low will give you some feel for how it proceeds. Note I have not attempted to be completely accurate to the algorithm

is animation, but simply to give you an intuitive feel for how it works. You will notice that on some steps the algorithm

ll fill in the labels in blue and on other steps it will fill in the labels in green. These latter steps, those that involve green

present the case where the label is simply propagated along a line from one vertex to the next.



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The point at which the algorithm begins is arbitrary. In this example figure, the algorithm would arrive at a unique

beling for each line in the figure. This is what typically happens, and it can happen quite rapidly because of the small

mber of possible labelings for any vertex and the way in which the labelings constrain each other because of the rigid

dy assumption mentioned earlier. Intuitively, what this assumption says is that if the world is made up of rigid objects,

en the shape of one object can't really depend on the shape of another object...there are no constraints to propagate

yond object boundaries. Note this is not the case with non-rigid objects such as flags or pools of water. Toss a rock into

pool of water, and the effect is propagated indefinitely.

We know from the study of dependency networks and the analysis of algorithms for propagation of dependency thatnerally this class of problems is computationally complex (or intractable to use a more technical term)....that is, if you

ve a large problem then it may take a very long time to see if there is a consistent way in which to assign values to the

twork of dependencies.

The algorithm works efficiently for partitioning a physical scene into objects because the rigid-body assumption hold

ost of the time. And, it appears that this knowledge is "built into" our way of reasoning about such scenes. Thus, this is

example where the mind is biased, but biased in a way that allows it to efficiently, and usually accurately, interpret

ysical scenes.

What about all those crazy pictures at the beginning? Well, what I was attempting to do was to create pictures that we

mplex enough to either defeat your ability to maintain focus on a coherent set of vertices or create pictures that were

mbiguous, that is, had more than one consistent labelings. The picture below depicts a "ribbon" of vertices. If you focusthe center, you will see that there are not enough constraints to force you to a single interpretation. You can see a line a

nvex or concave and it can shift back and forth. Your mind can give this picture more than one consistent interpretatio

There are some pictures that you can give no consistent interpretation. But your mind has a hard time determining

actly why. Many of the constraints hold "locally" but there is not a consistent global interpretation. These types of

gures are often referred to as impossible figures...because they are. The impossible triangle that was used earlier is

own again below on the left. It is also shown on the right but with circles used to impose "local" views of the figure. If

u look at each of these views, you will see that each is perfectly fine. It is when the constraints get "passed" to the next

ew that we recognize the impossibility of the figure.



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And, just in case this all hasn't seemed impossible enough....here are two more figures. The impossible prong on the

ght and an impossible nut on the left!


Charles F. Schmidt



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roblem Solving and Productive Thinking

Problem Solving - The Cryptarithmetic Example

In addition to research in perception, the Gestaltist also focused on problem solving. In retrospect, we can see that

roblem solving often exhibits two properties that support the Gestaltist position. First, it is typically the case that many

eps are required to solve a problem. These "steps" often don't need to be (and in some case couldn't be) made in a

hysical sense. This complexity of the "response" supports the idea of a mind actively manipulating "mental stuff."

econd, many times a problem can be represented in differing ways...and, the way in which the problem is represented

ften crucial to the solution.

In this section, we will use several different problems to exemplify and amplify these points. The first problem that w

ill examine is an example of a cryptarithmetic problem. The problem statement is given below. This same problem i

ven in the text. You should take some time trying to solve the problem so that our discussion of this problem will be

asier for you to grasp.

If you were unable to solve the problem, try to determine why. If you solved it, think back over your problem solvin

fforts and try to identify points where you ran into particular difficulties.

Problems such as this are not easy to solve. You must "think" of the problem in the right way, follow a strategy for

ursuing the solution, and be systematic in remembering intermediate steps in your solution.

One of the questions that we would like to understand better is exactly what makes one problem hard and another ea

n this introductory level course we won't be able to develop the sophistication required to fully describe the way this

uestion is being addressed, but we can begin to point out some of the features of problem solving that are importantly

lated to problem difficulty.

First, a little review. Previously, we discussed the line labeling problem. We noted that the general problem of

atisfying some arbitrary set of constraints over a set of objects is a very difficult problem. It is what is referred to as an

ntractable problem. But the physical world, it turns out, isn't really an arbitrary set of objects. Rigid objects can't

fluence other rigid objects over arbitrary distances. This fact, and just as importantly, the fact that our mind seems to

tilize this fact, makes the line labeling problem tractable.

Now, it turns out that this cryptarithmetic problem is also a problem that involves satisfying a set of constraints. But

our mind doesn't automatically exploit the implications of this fact. You must have some knowledge of algebra and us

is to construct an algebraic representation of the problem to actually take advantage of the fact that this is a constraint

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atisfaction problem.

Note, this is not the only way in which the problem could be represented. You could simply think of it as involving a

f 10 letters and a set of 10 integers and think of the basic problem solving move as assigning each of the integers to a

tter and testing to see if the assignment is correct. This set theoretic representation together with this move of assign

tegers to letters is, in fact, a way in which the problem can be solved. I know of no agent, human or machine, that eve

olved it in this fashion. The reason is that even with one of the assignments given, namely D=5, there are still 9 more

ssignments to make. And, there are 9! (362,880) ways of making these assignments. A very large set to look through!

nd, to make matters worse, there is no way to know when you are getting close. In this way of representing and thinki

bout the problem, an assignment is either correct or incorrect. There is no such thing as being partially correct.

But now contrast this set theoretic representation with the algebraic representation of the problem. This representa

depicted below. The representation consists of rewriting the problem as a set of six equations where we have explicit

dded the carry terms (c1, c2, ....) that are involved in the addition. At the top of the picture in the green box, the possib

alue assignments are explicitly shown (the 'v' is the symbol used to represent logical 'or'). The picture also includes arr

nking the terms in the equation. Each of these words is six letters long, and there are only 10 letters involved in the

roblem. Thus, letters must recur at various points in the equations. The arrows link places where the same letter occur

ow, we know that the integer that we assign to a letter in one equation constrains the integer that can be assigned to

nother letter in the equation. For example, assigning 5 to D in (1) constrains us to assign 0 to T and 1 to c1. If these let

ccur in other equations, then this assignment affects those equations....again, the constraints propagate!



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Note that once we have represented the problem as one that involves six subproblems, we now have the opportunity

ntelligently) choose which subproblem to work on first, which next, and so on. Intelligent choices involve following t

ependencies (as well as knowing a bit about how to exploit algebraic identities, as in equation 5) so that the number of

ossible candidate values for a particular letter is strongly constrained.

The figure below was created to help you visualize these constraints or dependencies between subproblems. The lett

present the letters in the problem. Each rectangle or triangle represents one of the six subproblems. The triangles

present the case where an equation involves three letters and the rectangles where only two letters are involved. The

tters within a shape and the line between them also serve to indicate that the letters occur in the same equation. Thetersection of the figures reflects those cases where the same letter appears in differing equations. This gives you an

verall picture of the structure of the dependencies in this problem. The picture to the right also includes the carry term

nd each geometric figure is labeled with the corresponding equation or subproblem.

By seeing the problem represented in this way, I hope that the relation between this type of problem solving and the

beling problem is now apparent. Think of each equation (the rectangle or triangle above) as corresponding to a vertex

he letters of the equation then correspond to the lines in the vertex, and the integers are the "labels" for the letters

orresponding to the labels for the lines.

We have seen two very different cases where a problem has been broken down or decomposed into parts orubproblems. And, having found a "good" decomposition and an order in which to work on the subproblems; the proble

ecomes rather easy to solve. This will be a recurring theme. But be warned, we can't always find a good decompositio

nd, there are a great many to look through. Recall the partitioning of the dots and the Sterling number for the number

artitions....the same combinatoric principles apply to problem decompositions.


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heckerboard Problem

Problem Solving and Problem Representation

Below is presented what is referred to as the checkerboard problem. An illustration of the starting state of this proble

so presented to help you think about the problem solution. Read the problem statement and solve the problem.

Now, that you have solved this problem, move on to a small variant of this problem called the mutilated checkerboaroblem.

Mutilated Checkerboard Problem Productive Thinking...the Gestalt Empha

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Mutilated Checkerboard Problem


Below is presented what is referred to as the mutilated checkerboard problem. An illustration of the starting state of

oblem is also presented to help think about the problem solution. Read the problem statement and solve the problem.

Were you able to solve this problem? If so, did it take awhile to get on track? Now move on and try the matchmaker

roblem.

Matchmaker Problem Productive Thinking...the Gestalt Emphas

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MatchMaker Problem


Below is presented what is referred to as the Matchmaker Problem. Read the problem statement and solve the proble

You should have found this very easy to solve. And, you probably have noticed that there is a abstract resemblance

etween this problem and the mutilated checkerboard problem. Hopefully, your experience with these problems has ale

ou to the importance of how a problem is thought about, that is, how a problem is represented.


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erms, Concepts, Questions

Some Terms, Concepts and Questions

Productive Thinking

Reproductive Thinking

What are some of the reasons for proposing

this distinction?

roblem RepresentationAlternatives and Choice

Shift or Change of Representation

What determines these choices?

What is the relation between Choice and

Bias?

Problem Decomposition

Problem Representation

Problem Constraints

Subproblem Dependencies

What makes a problem difficult?

Where do problem decompositions come

from?

Constraint Propagation

Dependence

Independence

Consistency

How is knowledge incorporated into this

procedure?

Productive Thinking...the Gestalt

Emphasis

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raffic Lights - Possibilities and Dependencies

Traffic Lights - Possibilities and Dependencies

Illustrated below is a "traffic light" ...a vertical arrangement of a red, an amber, and a green light each of which can b

her on or off. We will use the idea of traffic lights at an intersection to illustrate the relation between the concepts of

ssibilities, dependencies, and constraints. In addition to the basic traffic light, the illustration below lists two sets of

ssibilities. One is termed the logical possibilities (or in this case you could think of it as the physically logical

ssibilities). There are 8 logical possibilities ranging from the case where all three light are on at the same time to the chere all three lights are off at the same time. Note that each object - one of the lights - can be in one of two states - on o

f. Since there are three lights, the number of logical possibilities is the number of states raised to the power of the num

objects - in this case 2 to the 3.

In the darker gray box is the set of what I have termed 'designed possibilities.' This is the set of possibilities that any w

haved traffic light will exhibit. And there are only three such possibilities, the cases where one of the three lights is on

e remaining are off. Finally, the pointers to the circles is meant to convey the obvious, but important fact that the desig

ssibilities are a subset of the logical possibilities.

Keeping these ideas in mind, pretend that for some reason you can't see a tr

light, but your companion traveling with you can. Further pretend that when yask your companion to tell you the state of the traffic light, your companion sa

that it is red. Notice that if you assume that this is a well-behaved traffic light,

you know all you need to know because the only designed possibility in which

red light is on is one where the green and yellow lights are off. You can "infer

because this dependency between the on\off values of the three lights will alw

hold for well-behaved traffic lights. Whenever, the value of one thing depends

the value of another we say that there is a dependency between these two thin

Notice that in some cases the dependency may be such that knowing one value

uniquely determines another value. That is the case here...knowing that the red

light is on uniquely determines the value of the remaining lights. However,

knowing that the yellow light is off doesn't allow us to infer the value of the grlight or the value of the red light. It does, however, let us infer that one of thes

lights is on and one is off.

Anytime, there is a dependency among the values that different entities can

take, we can potentially take advantage of these dependencies to reason about

objects. Working cross word puzzles is one long exercise in attempting to use

dependencies to reason to a unique set of values for each square in the puzzle.

Now let us gain some experience in explicitly thinking about possibilities and dependencies. Consider the standard tr

ersection with a light at each of the four corners. Work out the number of logically possible values that this four light

nfiguration can take on. Next work out the 'designed' possibilities and write out the inferences that can be derived whe

ese dependencies hold. Finally, work out a system that functions in the same manner but now uses only two lights ... ad a green light on each "traffic light." Finally, decide whether you could reduce the number of lights use to just one...s

een light on each "traffic light."


Charles F. Schmidt

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ONALD + GERALD = ROBERT

The Cryptarithmetic Problem

The problem below is an example of a cryptarithmetic problem. The problem statement is given below. This sam

oblem is given in your text. Solve this problem and try to write down as detailed a record as possible of your thinkin

nd decisions as you attempt to solve this problem.

If you were unable to solve the problem, try to determine why. If you solved it, think back over your problem solvi

forts and try to identify points where you ran into particular difficulties.


Charles F. Schmidt

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Matchstick Problems - Table of Contents

MatchStick Problems

q Four Squares

q Four Squares Solutions

q 40 Squares Problems

q Large and Small Matches Problems


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able of Contents 305

o to Course Syllabus

Table of Contents

art I.

istorical Perspective and Basic Approaches to the Study of Thinking

q Introduction

q Associationism and Behaviorism

q Productive Thinking...the Gestalt Emphasis

q Experimental Decomposition of Thinking

q Computational Approach to the Study of Thinking

q Cognitive Development and Learnability

art II.

spects of Thinking/Cognition

q Deduction

q Induction, Concepts, and Reasoning under Uncertainty

q Understanding, Interpreting and Remembering Events

q Problem Solving and Planning

Home Course Materials Page

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ntroduction Contents

ntroduction

Lecture Material

q Some Quotes

q The Mind and Formal Systems

r Some Examplesq Structure and Randomness Discussion

r Media and Memory

r Structure Experiment

Terms, Concepts, and Questions!

q Terms and Concepts, and Questions


q Exercise/Assignment ...What Kind of Mind..

q Exercise/Assignment ...What does the Mind See? Some views...

q Exercise/Assignment ...Looking for Structure in Recall

q Exercise/Assignment ...What is Thinking and what isn't

q Exercise/Assignment ... What kind of memory is a photograph?

Table of Contents

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yllabus 305

Syllabus 830:305 Cognition - Section 1

Fall, 2005

Text: Mayer, R. E. Thinking, Problem Solving, Cognition. Second Edition. New Y

W. H. Freeman, 1992.

Place: Room PH 115, Busch Campus

Time: Monday & Thursday 1st Period (8:40 - 10:00 A.M.)

Instructor: Prof. Charles Schmidt

Office: Room 135A, Psychology Bldg, Busch Campus

Phone: 732 445-2874

Email: [email protected]

Course URL: http://www.rci.rutgers.edu/~cfs

Office Hours: Monday, 12:00 - 1:00 PM or by appointment

T.A.

lick here for PDF version of the Syllabus

Course Outline

Part I. Historical Perspectives and Basic Approaches to the Study of Thinking

Introduction

Reading:Chapter 1. Beginnings. pp 3-18.

Website: Introduction

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yllabus 305

Associationism and Behaviorism

Reading:Chapter 2. Associationism: Thinking as Learning by Reinforcement. pp. 19-38.

Website: Associationism and Behaviorism


Reading:Chapter 3. Gestalt: Thinking as Restructuring Problems. pp. 39-78.

Website: Productive Thinking...the Gestalt Emphasis

Experimental Decomposition of Thinking

Reading: Chapter 7. Mental Chronometry: Thinking as a Series of Mental Operations. pp. 203-224

Website: Experimental Decomposition of Thinking

EXAM 1

Computational Approach to the Study of Thinking

Reading:Chapter 6. Computer Simulation: Thinking as a Search for a Solution Path. pp. 167-202.

Website:Computational Approach to the Study of Thinking

Cognitive Development and Learnability

Reading:Chapter 10. Cognitive Development: Thinking as Influenced by Growth. pp. 283-323.

Website:

Part II. Aspects of Thinking

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yllabus 305

Deduction

Reading: Chapter 5. Deductive Reasoning: Thinking as Logically Drawing Conclusions. pp. 283-323.

Website: Deduction

EXAM 2

Induction, Concepts, and Reasoning under Uncertainty

Reading: Chapter 4. Inductive Reasoning: Thinking as Hypothesis Testing. pp. 81-113.

Chapter 9. Question Answering: Thinking as a Search of Semantic Memory. pp. 259-279

Website: Induction, Concepts, and Reasoning under Uncertainty

Understanding, Interpreting and Remembering Events

Reading:Chapter 8. Schema Theory: Thinking as an Effort after Meaning. pp. 225-258.

Website: Understanding, Interpreting and Remembering Events

Problem Solving and Planning

Reading: Chapter 14. Analogical Reasoning:Thinking as Based on Analogs, Models and Examples. pp. 415-454.

Chapter 13. Expert Problem Solving: Thinking as Influenced by Experience. pp. 387-414.

Chapter 15. Mathematical Problem Solving: Thinking as Based on Domain-Specific Knowledge. pp.

455-489.Chapter 16. Everyday Thinking: Thinking as Based on Social Contexts. pp. 490-507.

Website: Problem Solving and Planning

Final Exam (Dec 17 8-11AM)

Homework, Grading, Etc.



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yllabus 305

Exams

The exams will cover the material presented in the text, lectures, and the website.

Assignments

The website will often include for a section some assignments, exercises or questions to be considered. These activities

are primarily intended to focus your thinking about the course material being covered. In many cases there may be no

obviously correct answer. In other instances, the primary purpose of the exercise is to help you to reflect upon your ownthinking and performance when doing a cognitive task. The assignments may be discussed in class.

Class Participation

The lecture material for much of this course is provided on the website. The purpose of this is not to relieve you of the

onerous task of attending class. The purpose is to:

q Allow you to read over the material before it will be presented and discussed in class so that you can determine

which aspects of the material you may not understand;

q Relieve you of the necessity of taking extensive notes in class and giving you the freedom to follow the lecture a

discussion actively and critically;q Relieve me from covering in detail the material, allowing me to emphasize the main points of the material as we

as add additional information;

q Provide additional time and opportunity for questions and discussion in class;

q Finally, you will notice that the material in the website often includes examples of the ideas under discussion. It

important for you to not only work through the examples, but also to make sure that you understand the basic ide

or concept that the example is being used to illustrate. The examples will, at times, include a great deal of detail

since this can easily be included in the web pages. The detail is there to help you develop your intuitions

concerning the ideas presented. It is not presented as something that you are expected to be able to reproduce.

If at all possible, I suggest that you at least skim the material on the website prior to the class in which the material will

presented. If there are aspects that you do not understand but you are reluctant to ask about it in class, then you might

want to let me know about this via Email ([email protected]) prior to the class in which the material will be presented

will make it a point to read this directory prior to class.

Extra Credit

If you wish to do an extra credit project for the course, then this project should be approved no later than the thirdt week

in October and turned in to the instructor no later than Monday, Dec. 12. Some possible extra credit projects include:

q creating content related to the course that could potentially be included in the course website. This might involve

creating examples or experiments related to the course material, extending either the depth with which a topic iscovered, or adding additional related topics.

q creating additional tools for use of the website.

q participating in some research on problem solving. In this case, you would analyze and discuss your own data in

relation to various ideas about human problem solving.

q ...

Course Grade

Your course grade will mainly be determined by your performance on the exams. In addition to these exams, your

participation in class - questions formulated, discussion, and assignments - may also contribute to the determination of

mailto:[email protected]:[email protected]


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yllabus 305

your grade for the course. And, of course, any extra credit work will also be considered when assigning the final course

grade.

Printing the pages on this site

If for some reason you decide that you wish to print one or more of these pages then be sure that the print setup is in

landscape mode. Note, however, that these HTML pages have not been constrained to have any particular vertical limi

Consequently, a page may print onto several pages and the page breaks may occur at arbitrary points. If at all possible, Irecommend making every effort to use these pages on line rather than printing since they were developed under the

assumption that this would be the primary mode of use. Using them on line will allow you to view the animations,

JavaScript related features as well as view the most recent updates of the pages.

Table of Contents

Charles F. Schmidt



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Lecture Material

q Some Quotes

q Some S-R Theory




q Anagram Exercise/Assignment

r Anagram Critique

Table of Contents

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Mental Chronometry

Experimental Decomposition of Mental Processes

Lecture Material

q Mental Olympics

q Scanning Short Term Memory

q Additive Factors and Analysis of Variance


q Some Terms, Concepts and Questions


q What Might be Predicted ....?

Table of Contents

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OC: Computational Approach

Computational Approach to the Study of Thinking

Lecture Material

q Following Instructions

q Machines/Automata

q Levels Hypothesisq Search / Generate and Test

q Search Control Strategies

q Search Control Animations

q Problem Reduction Search

q Production Rule Example




q Computation and Computers as Physical Devices

q Computing and Thinking

Table of Contents

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eduction

Deduction

Lecture Material

q Propositional Logic:Some Intuitive Ideas

q Reasoning in the Syntax and in the Semantics; An Example

q Contradition and Proofq Euler Diagrams and Quantified Expressions

q Defeasible Inference: Inheritance

q Deduction Overview

Reference Material

q Truth Tables

q Some Logical Identitites

q Some Logical Implicationsq Liars Paradox

q Some Definitions for First Order Logic

q Some Rules for Quantifiers

q Some Definitions of Terms used in the Study of Formal Systems

q Resolution Theorem Proving




q Describing Things

Table of Contents

Charles F. Schmidt

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nduction, Concepts, and Reasoning under Uncertainty

Induction, Concepts, and Reasoning under Uncertainty*

Lecture Material

q Introduction

q Knowing a Concept and Concept Indistinguishability

q Example of Concept Identification Taskr Possible Hypotheses after One Example

q Structuring the Hypotheses Space and Hypothesis Revision

r Version Space and Learning the Concept of an Arch

q Example of Inducing Rules on Patterns

q Standard Probability Axioms and Beliefs

q Belief Revision/Bayes Rule

q The Grue Property

Reference Material

q Algebra of Sets and Definition of a Lattice

q Graph of a 3 Element Lattice

q Dutch Book

q http://www.pbs.org/wgbh/pages/frontline/shows/gamble/




Table of Contents

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Lecture Material

q Properties of Language

q Syntax and Sentence Understanding

q Parsing Sentencesq NonDeterminism and Parsing

q CaseGrammar

q Example of Representing Textual Information

q Example of Story Interpretation




Table of Contents

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Problem Solving and Planning

Lecture Material

q River Crossing Problems

r Missionary and Cannibals

r Jealous Husbandsr Solution Spaces

q Tower of Hanoi

r Tower of Hanoi Story

r Tower of Hanoi 3 Disk Solution

r The Tower of Hanoi 3 - Disk Space

r Tower of Hanoi Problem Decomposition

r n Disks and m Pegs "Tower of Hanoi" Example Problem

r A Problem from Tower of Hanoi Space

r Monster Problems

r Related Spaces


Terms, Concepts, Questions - Problem Solving


Table of Contents

Charles F. Schmidt

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ourse Materials for Courses Taught By Prof. Charles F. Schmidt

Syllabi and Course Materials for:

Fall, 2005 830:472 Computation and Cognition

Fall, 2005 830:305 Cognition

These materials are under active development. They are intended for us

students at Rutgers taking one of the above courses with Prof. Schmidt

Consequently, no explicit attempt has been made to insure that these

materials form a coherent presentation of the material for someone who

not taking the associated course from this instructor. As you might exp

there is considerable overlap in the material covered in these courses. T

course 830:305 Cognition is a general introduction to the study of hum

cognition; whereas the course 830:472 focuses on the study of cognitio

(by human or machine) as a kind of computation. Consequently, 472 go

into the relation between computation and cognition more rigorously,

deeply and technically than is done in 305.

In general, the materials for each course are different although in some cases the same material is used in each site. W

ere is a corresponding section in each course, you may find it helpful to consult that section in the course that you are

king. I have tried to keep the presentation of the materials in the 305 course at as simple and intuitive a level as the ma

ows. Consequently, if you a student in 472 and are having difficulty with the material in that course site, you might

nsult the corresponding section in the 305 site. Or if you are bored to tears with the simplicity of the material presente

e 305 site, you might see if there is a corresponding section in the 472 site.

These materials are constantly under development. Some sections are incomplete. Inquiries, comments, corrections, encerning these materials may be sent to:

Prof. Charles F. Schmidt

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mailto:[email protected]:[email protected]


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uotes

Some Quotes Exemplifying the Rationalist and Empiricist Positions *

Here are some quotes that highlight the rationalist and empiricist views of the human mind. Leibniz and Boole illust

e rationalist position where the mind is regarded as an entity that, like a mathematical system, follows rules that are un

itself and in a sense independent of the external world. Locke and Hume, speaking from the empiricist position, do no

e mind as independent, but as derivative from the world of experience.

During the first half of the 20th century, the empiricist position dominated the thinking and research on human reaso

the United States. During the second half of the century, the argument against this position intensified and the rationa

sition has achieved increasing dominance. Much of the impetus for the rationalist position has arisen from the

athematics associated with defining and studying computation as well as from the experience of using computation in

eryday life.

The quotes below provide only glimpse of these individuals thoughts and works. Where possible I have provided a lin

online version of the original text or to a related writing in case you wish to pursue their ideas further. )

Gottfried Wilhelm von Leibniz (1646-1716), On reasoning. 1677

"All our reasoning is nothing but the joining and substituting of characters, whether these

characters be words or symbols or pictures, ... if we could find characters or signs appropriate for

expressing all our thoughts as definitely and as exactly as arithmetic expresses numbers or

geometric analysis expresses lines, we could in all subjects in so far as they are amenable to

reasoning accomplish what is done in Arithmetic and Geometry.

For all inquiries which depend on reasoning would be performed by the transposition of

characters and by a kind of calculus, which would immediately facilitate the discovery of beautifuresults ..."

George Boole (1815 - 1864),An Investigation of the Laws of Thought on which are Founded theMathematical Theories of Logic and Probabilities (London, 1854);

"Nature and Design of this Work

The design of the following treatise is to investigate the fundamental laws of those operations of

the mind by which reasoning is performed; to give expression to them in the symbolical languag

of a Calculus, and upon this foundation to establish the science of Logic and construct its metho

to make that method itself the basis of a general method for the application of the mathematical

doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to vie

in the course of these inquiries some probable intimations concerning the nature and constitution

the human mind."

The Calculus of Logic by George Boole, first published in The Cambridge and Dublin Mathematical Journal, vol. 3

1848)

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uotes

ohn Locke (1632-1704) on Empiricism fromEssay concerning human understanding(1690):

"2. All ideas come from sensation or reflection. Let us then suppose the mind to be, as we say,

white paper, void of all characters, without any ideas:- How comes it to be furnished? Whence

comes it by that vast store which the busy and boundless fancy of man has painted on it with an

almost endless variety? Whence has it all the materials of reason and knowledge? To this I

answer, in one word, from EXPERIENCE. In that all our knowledge is founded; and from that it

ultimately derives itself. Our observation employed either, about external sensible objects, or

about the internal operations of our minds perceived and reflected on by ourselves, is that which

supplies our understandings with all the materials of thinking. These two are the fountains of

knowledge, from whence all the ideas we have, or can naturally have, do spring."

David Hume (1711-1776) fromAn Enquiry Concerning Human Understanding (1777 edition)

"Nothing, at first view, may seem more unbounded than the thought of man, which not only

escapes all human power and authority, but is not even restrained within the limits of nature an

reality. To form monsters, and join incongruous shapes and appearances, costs the imagination

no more trouble than to conceive the most natural and familiar objects. And while the body is

confined to one planet, along which it creeps with pain and difficulty; the thought can in an

instant transport us into the most distant regions of the universe; or even beyond the universe,into the unbounded chaos, where nature is supposed to lie in total confusion. What never was

seen, or heard of, may yet be conceived; nor is any thing beyond the power of thought, except

what implies an absolute contradiction.

But though our thought seems to possess this unbounded liberty, we shall find, upon a nearer examination, that it is

ally confined within very narrow limits, and that all this creative power of the mind amounts to no more than the

culty of compounding, transposing, augmenting, or diminishing the materials afforded us by the senses and experienc

hen we think of a golden mountain, we only join two consistent ideas, gold, and mountain, with which we were

rmerly acquainted. A virtuous horse we can conceive; because, from our own feeling, we can conceive virtue; and thi

e may unite to the figure and shape of a horse, which is an animal familiar to us. In short, all the materials of thinking

e derived either from our outward or inward sentiment: The mixture and composition of these belongs alone to the

ind and will. Or, to express myself in philosophical language, all our ideas or more feeble perceptions are copies of ou

mpressions or more lively ones.To prove this, the two following arguments will, I hope, be sufficient. First, when we

alyse our thoughts or ideas, however compounded or sublime, we always find, that they resolve themselves into such

mple ideas as were copied from a precedent feeling or sentiment. Even those ideas, which, at first view, seem the mos

de of this origin, are found, upon a nearer scrutiny, to be derived from it. The idea of God, as meaning an infinitely

telligent, wise, and good Being, arises from reflecting on the operations of our own mind, and augmenting, without

mit, those qualities of goodness and wisdom. We may prosecute this enquiry to what length we please; where we shall

ways find, that every idea which we examine is copied from a similar impression."

IntroductionCharles F. Schmidt

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ormal Systems and a Language of Thought

Formal Systems and a Language of Thought

"All our reasoning is nothing but the joining and substituting of characters, whether these characters be words

or symbols or pictures, ... if we could find characters or signs appropriate for expressing all out thoughts as

definitely and as exactly as arithmetic expresses numbers or geometric analysis expresses lines, we could in all

subjects in so far as they are amenable to reasoning accomplish what is done in Arithmetic and Geometry."

Leibniz (1677)

Judging from the quote above, it appears that Leibniz was quite certain that thinking of the mind as a formal system

a useful way to view reasoning. The explicit idea of a formal system is pretty much an intellectual product of this

century, but Leibniz uses areas of mathematics as examples of what he has in mind. These areas certainly qualify as

examples of what we would now refer to as a formal system.

The interest in the idea of a formal system arises from the intuition that there is a kind of "language of thought."

ndeed, a naive assumption is that the language of thought is determined by the natural language that we have learned

as a child. And, certainly it is hard to escape the intuition that the language we speak is intimately related to our thou

But we needn't resolve this issue now, because in this century the idea of a formal language or system has been well-

defined. In one sense, this is simply an abstraction of the idea of a natural language. And, as such it provides a clearpresentation of some of the basic properties of a "language." We could be very careful and exact in defining the class

hings that we call formal systems, but at this point we just want to get the idea out there so you can use it to help thi

about the issues that were, and still are argued about, by people who study human reasoning.

A formal system consists first of all of a set of things, usually we think of this set of things as a set of symbols. A

ymbol is something that someone "dreams up" as opposed to something that nature provides on its own. To capture

his idea, it is often said that symbols are 'arbitrary.' For example, the letter 'A' is not a phenomenon of nature...

omeone decided to adopt this set of conventions to make this form and treat it as - the letter 'A'. And, a symbol does

automatically refer to anything other than itself...you and I had to learn that the letter 'A' could be used to refer to the

ound-A. Another property of symbols is that we usually try (or are taught to try) to make the symbols unambiguou

f you are writing an 'A' you try to write it in such a way that it won't be confused with any other symbol that is in the

et of symbols you are using.

The numbers used in mathematics, letters used to write down a natural language, notes used to write down music, are

ust some of the familiar examples of differing sets of symbols. Notice that the letters of our alphabet and the notes

used in music are each finite in number...there are 26 letters in the English alphabet. But we can use these finite sets

create sets of things, expressions; and the set of possible expressions is not really bounded in size....the set is infinite

n a technical sense, there are an infinite number of sentences in the English language and we can use the alphabet to

express each of these. A similar claim could be made about the number of musical expressions.

In the picture on the left, I have used three different sets o

symbols - integers {3,2,5...}, a 'stick' {|}, and the English

alphabet {B,a,n,...}. In each of the gray boxes I have groupthese symbols in particular ways to serve as examples for t

discussion. First, note that I added some symbols - for

example, + and = as well as a blank space and and period (

in the case of the sentences.

These symbols seem to be a bit different and they are. Rec

that we have only a finite number of symbols but we want

be able to create an infinite set of things that we call

expressions from this finite set. Well, the only way in whic

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to obtain an infinite set of expressions from a finite set of

expressions is to define ways in which to compose

expressions from the elements of the vocabulary. +, = and

space in the case of sentences are used to represent a

composition of elements of a set. For example,

2 is a number (expression)

3 is a number (expression)

2 + 3 is a number (expression)

2 + 3 + 2 is a number (expression)

2 + 3 + 2 + 3 is a number (expression)

and so on.

So how does this help us think about the mind. Well, perha

the mind has a finite vocabulary of "basic ideas"....perhaps

has a finite set of ways of composing these ideas into well-

formed expressions (complex ideas)...and perhaps it has a of syntactically defined rules of inference. Perhaps, then th

is a sense in which we have an infinite set of ideas (how do

we fit them into our brain then?) And, perhaps the mind ca

imagine syntactic expressions that are false or describe a

completely imaginary world such as Alice's Wonderland.

Could this be possible without a language of thought?

But as soon as we allow ourselves to string elements of our set together, we need to introduce the idea of following

ules for stringing them together. We call these rules, syntactic rules....they are rules that define the way in which w

orm expressions using our basic set of symbols. In the figure above, the first two sentences in the lower box are

yntactically correct. The last sentence, shown in red is not syntactically correct. A more general term that is often us

o refer to this distinction is to say that syntactically correct expressions are well-formed expressions or formulae.

Now we can create an infinite set of expressions from a finite set of elements. Can there be more? Well, yes. We wou

ike to able to say something more about these expressions...more specifically, we would like to be able to say

omething about possible relations between elements of these expressions. Note, that I have exemplified the

commutative law" in the equations in the upper right. Now, if the commutative law holds; then if we have the

expression '2 + 3 = 5,' then we can infer or derive the expression '3 + 2 = 5' using the commutative axiom. This

epresents a rule of inference and rules of inference are another component of a formal system. I used a similar type

ule with the "Bacon and Eggs" phrase to derive the lower sentence from the first. Note that the rule of inference say

omething about how to modify one expression to yield another...and technically, that is all it says.

This last point is important ... formal systems are also often called syntactic systems to contrast them with systemswhere the expressions are intended to refer to something outside the system...to have an associated semantics.

Now, this can get really tricky but the intuitions are familiar. "Bacon and Eggs have high Cholesterol." is simply a w

ormed expression and nothing more from the syntactic point of view. But, of course, these words refer to something

outside the syntactic system and in addition to being syntactically correct, the sentence may be semantically correct..

may make a true statement about the things that the word 'Bacon' and the word 'Eggs' and the word 'Cholesterol' refe

n the world.



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So how does this help us think about the mind. Well, perhaps the mind has a finite vocabulary of "basic ideas"....

perhaps, it has a finite set of ways of composing these ideas into well-formed expressions (complex ideas)...and perh

t has a set of syntactically defined rules of inference. Perhaps, then there is a sense in which we have an infinite set o

deas (how do we fit them into our brain then?) And, perhaps the mind can imagine syntactic expressions that are fal

or describe a completely imaginary world such as Alice's Wonderland. Could this be possible without a language of

hought?

This idea that the mind possesses a "language of thought" in this formal sense is, more or less, the rationalist positio

This stands in contrast to the empiricist position that relies on the "world outside our mind" to populate our mind wit

deas.

Introduction

Charles F. Schmidt



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ome Examples

Some Examples

his page provides some examples that may help you to understand the idea of a combinatoric system and that of a

rmal system.

formal system consists of:

q A finite set of symbols or vocabulary, .

q A set expressions that are formed from the vocabulary. This set of expressions is a subset of

q Rules of inference.

r example, if the vocabulary is the set of words:

{ Jim, is, old }

en the set ofpossibleexpressions that could be formed using concatenation to combine (i.e, putting one element

er another) the elements of this vocabulary are:

{Jim, is, old,

Jim Jim, Jim is, Jim old,

is Jim, is is, is old,

old Jim, old is, old old,

Jim Jim Jim, Jim is Jim, Jim old Jim,

is Jim Jim, is is Jim, is old Jim,

old Jim Jim, old is Jim, old old Jim,Jim Jim is, Jim is is, Jim old is,

is Jim is, is is is, is old is,

old Jim is, old is is, old old is,

Jim Jim old, Jim is old, Jim old old,

is Jim old, is is old, is old old,

old Jim old, old is old, old old old

...}

he set ofpossible expressions is referred to as and is simply the set of all combinations of the elements of thecabulary of length n where n = 1, 2, 3, 4, ... We have explicitly shown the sets of possible expressions of length 1, 2

d 3 for this three element vocabulary. There are 3 expressions of length 1; 9 of length 2, and 27 of length 3. In

neral, if the vocabulary has n elements then the total number of possible expressions of length j or less is given by th

mmation over j of n to the jth power. Thus, for this example, the total would be 39 for j =3, 120 for j = 4, 363 for j =

1092 for j = 6 and so on.

his is an example of what is called a combi

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