1969

Turing
Award
Lecture

Form and Content

in Computer Science
MARVIN MINSKY
Massachusetts Institute of Technology
Cambridge, Massachusetts

An excessive preoccupation with formalism is impeding the development of

computer science. Form-content confusion is discussed relative to three areas: theory
of computation, programming languages, and education.

The trouble with computer science today is an obsessive concern

with form instead of content.
No, that is the wrong way to begin. By any previous standard the
vitality of computer science is enormous; whaf other intellectual area
ever advanced so far in twenty years? Besides, the theory of computa-
tion perhaps encloses, in some way, the science of form, so that the
concern is not so badly misplaced. Still, I will argue that an excessive
preoccupation with formalism is impeding our development.
Before entering the discussion proper, I want to record the satisfac-
tion my colleagues, students, and I derive from this Turing award. The
cluster of questions, once philosophical but now scientific, surrounding
the understanding of intelligence was of paramount concern to Alan
Turing, and he along with a few other thinkers--notably Warren S.
McCulloch and his young associate, Walter Pitts -- made many of the

Author's present address: MIT Artificial Intelligence Laboratory; 545 Technology Drive,
Cambridge, MA 021390.

219
early analyses that led both to the computer itself and to the new
technology of artificial intelligence. In recognizing this area, this award
should focus attention on other work of my own scientific family--
especially Ray Solomonoff, Oliver Selfridge, John McCarthy, Allen
Newell, Herbert Simon, and Seymour Papert, my closest associates in
a decade of work. Papert's views pervade this essay.
This essay has three parts, suggesting form-content confusion in
theory of computation, in programming languages, and in education.

1
Theory
of Computation
To build a theory, one needs to know a lot about the basic phenom-
ena of the subject matter. We simply do not know enough about these,
in the theory of computation, to teach the subject very abstractly.
Instead, we ought to teach more about the particular examples we now
understand thoroughly, and hope that from this we will be able to
guess and prove more general principles. I am not saying this just to
be conservative about things probably true that haven't been proved
yet. I think that many of our beliefs that seem to be common sense
are false. We have bad misconceptions about the possible exchanges
between time and memory, trade-offs between time and program
complexity, software and hardware, digital and analog circuits, serial
and parallel computations, associative and addressed memory, and
SO o n .
It is instructive to consider the analogy with physics, in which
one can organize much of the basic knowledge as a collection of
rather compact conservation laws. This, of course, is just one kind
of description; one could use differential equations, minimum prin-
ciples, equilibrium laws, etc. Conservation of energy, for example,
can be interpreted as defining exchanges between various forms of
potential and kinetic energies, such as between height and velocity
squared, or between temperature and pressure-volume. One can base
a development of quantum theory on a trade-off between certainties
of position and momentum, or between time and energy. There is
nothing extraordinary about this; any equation with reasonably smooth
solutions can be regarded as defining some kind of a trade-off among
its variable quantities. But there are many ways to formulate things
and it is risky to become too attached to one particular form or law
and come to believe that it is the real basic principle. See Feynman's
[1] dissertation on this.
Nonetheless, the recognition of exchanges is often the conception
of a science, if quantifying them is its birth. What do we have, in
the computation field, of this character? In the theory of recursive

220 MARVIN MINSKY

functions, we have the observation by Shannon [2] that any ~hring I 9 ti 9
machine with Q states and R symbols is equivalent to one with 2 'lurh~g
/lll'il I II
states and nQR symbols, and to one with 2 symbols and n'QR states, 1,1"¢' I lii't'
w h e r e n and n' are small numbers. Thus the state-symbol product QR
has an almost invariant quality in classifying machines. Unfortunately,
one cannot identify the product with a useful measure of machine
complexity because this, in turn, has a trade-off with the complexity
of the encoding process for the m a c h i n e s - - a n d that trade-off seems
too inscrutable for useful application.
Let us consider a more elementary, but still puzzling, trade-off,
that b e t w e e n addition and multiplication. H o w m a n y multiplications
does it take to evaluate the 3 × 3 determinant? If we write out the
expansion as six triple-products, we need twelve multiplications. If
we collect factors, using the distributive law, this reduces to nine.
What is the m i n i m u m number, and how does one prove it, in this and
in the n x n case? The important point is not that we need the answer.
It is that we do not know how to tell or prove that proposed answers
are correct! For a particular formula, one could perhaps use some
sort of exhaustive search, but that wouldn't establish a general rule.
One of our prime research goals should be to develop methods to
prove that particular procedures are computationally minimal, in
various senses.
A startling discovery was made about multiplication itself in the
thesis of Cook [3], which uses a result of Toom; it is discussed in Knuth
[4]. Consider the ordinary algorithm for multiplying decimal numbers:
for two n-digit numbers this employs n 2 one-digit products. It is usually
supposed that this is minimal. But suppose we write the n u m b e r s in
two halves, so that the product is N = (@ A + B) (@ C + D), where @
stands for multiplying by 10 "/2. (The left-shift operation is considered
to have negligible cost.) T h e n one can verify that

N -= @@AC + BD + @(A + B) (C +D) - @(AC + BD).

This involves only three half-length multiplications, instead of the

four that one might suppose were needed. For large n, the reduction
can obviously be reapplied over and over to the smaller numbers. The
price is a growing n u m b e r of additions. By compounding this and other
ideas, Cook showed that for any E and large enough n, multiplication
requires less than n 1+~ products, instead of the expected n 2. Similarly,
V. Strassen showed recently that to multiply two m x m matrices, the
number of products could be reduced to the order of m 1°g27 w h e n it was
always believed that the n u m b e r must be c u b i c - - b e c a u s e there are m 2
terms in the result and each would seem to need a separate inner

Form and Content in Computer Science 221

product with m multiplications. In both cases ordinary intuition has
been wrong for a long time, so wrong that apparently no one looked
for better methods. We still do not have a set of proof methods ade-
quate for establishing exactly what is the m i n i m u m trade-off exchange,
in the matrix case, between multiplying and adding.
The multiply-add exchange m a y not seem vitally important in
itself, but if we cannot thoroughly understand something so simple,
we can expect serious trouble with anything more complicated.
Consider another trade-off, that between m e m o r y size and computa-
tion time. In our book [5], Papert and I have posed a simple question:
given an arbitrary collection of n-bit words, h o w m a n y references to
m e m o r y are required to tell which of those words is nearest 1 {in n u m b e r
of bits that agree) to an arbitrary given word? Since there are m a n y
ways to encode the "library" collection, some using more m e m o r y than
others, the question stated more precisely is: h o w must the m e m o r y
size grow to achieve a given reduction in the n u m b e r of m e m o r y
references? This m u c h is trivial: if m e m o r y is large enough, only one
reference is required, for we can use the question itself as address, and
store the answer in the register so addressed. But if the m e m o r y is just
large enough to store the information in the library, then one has to
search all of i t - - a n d we do not know any intermediate results of any value.
It is surely a fundamental theoretical problem of information retrieval,
yet no one seems to have any idea about h o w to set a good lower bound
on this basic trade-off.
Another is the seria]~parallel exchange. Suppose that we had n
computers instead of just one. H o w m u c h can we speed up what kinds
of calculations? For some, we can surely gain a factor of n. But these
are rare. For others, we can gain log n, but it is hard to find any or
to prove what are their properties. And for most, I think, we can
gain hardly anything; this is the case in w h i c h there are m a n y highly
b r a n c h e d conditionals, so that look-ahead on possible branches will
usually be wasted. We k n o w almost nothing about this; most people
think, with surely incorrect optimism, that parallelism is usually a
profitable way to speed up most computations.
These are just a few of the poorly understood questions about
computational trade-offs. There is no space to discuss others, such as
the digital-analogquestion. {Some problems about local versus global
computations are outlined in [5].) And we k n o w very little about trades
b e t w e e n numerical and symbolic calculations.
There is, in today's c o m p u t e r science curricula, very little attention
to what is k n o w n about such questions; almost all their time is devoted
to formal classifications of syntactic language types, defeatist un-
solvability theories, folklore about systems programming, and generally
trivial fragments of "optimization of logic d e s i g n " - - t h e latter often in
For identifying an exact match, one can use hash-coding and the problem is reasonably
well understood.

222 MARVIN MINSKY

situations where the art of heuristic programming has far outreached I ~) (i ~)
the special-case "theories" so grimly taught and tested--and invocations 'luring
Aw~lrd
about programming style almost sure to be outmoded before the student [,e¢:ltlre
graduates. Even the most seemingly abstract courses on recursive
function theory and formal logic seem to ignore the few known useful
results on proving facts about compilers or equivalence of programs.
Most courses treat the results of work in artificial intelligence, some
now fifteen years old, as a peripheral collection of special applications,
whereas they in fact represent one of the largest bodies of empirical
and theoretical exploration of real computational questions. Until all
this preoccupation with form is replaced by attention to the substan-
tial issues in computation, a young student might be well advised
to avoid much of the computer science curricula, learn to program,
acquire as much mathematics and other science as he can, and study the
current literature in artificial intelligence, complexity, and optimization
theories.

2
Programming
Languages
Even in the field of programming languages and compilers, there
is too much concern with form. I say "even" because one might
feel that this is one area in which form ought to be the chief concern.
But let us consider two assertions: {1) languages are getting so they
have too much syntax, and (21 languages are being described with
too much syntax.
Compilers are not concerned enough with the meanings of expres-
sions, assertions, and descriptions. The use of context-free grammars
for describing fragments of languages led to important advances in
uniformity, both in specification and in implementation. But although
this works well in simple cases, attempts to use it may be retarding
development in more complicated areas. There are serious problems
in using grammars to describe self-modifying or self-extending
languages that involve execution, as well as specifying, processes.
One cannot describe syntactically--that is, statically- the valid expres-
sions of a language that is changing. Syntax extension mechanisms
must be described, to be sure, but if these are given in terms of a
modern pattern-matching language such as SNOBOL, CONVERT [6],
or MATCHLESS[7], there need be no distinction between the parsing
program and the language description itself. Computer languages
of the future will be more concerned with goals and less with pro-
cedures specified by the programmer. The following arguments are
a little on the extreme side but, in view of today's preoccupation with
form, this overstepping will do no harm. {Some of the ideas are due
to C. Hewitt and T. Winograd.]

Form and Content in Computer Science 223

 2.1
Syntax
Is Often Unnecessary
One can survive with much less syntax than is generally realized.
Much of programming syntax is concerned with suppression of paren-
theses or with emphasis of scope markers. There are alternatives
that have been much underused.
Please do not think I am against the use, at the human interface,
of such devices as infixes and operator precedence. They have their
place. But their importance to computer science as a whole has been
so exaggerated that it is beginning to corrupt the youth.
Consider the familiar algorithm for the square root, as it might
be written in a modern algebraic language, ignoring such matters as
declarations of data types. One asks for the square root of A,
given an initial estimate X and an error limit E.

DEFINE SQRT(A, X,E):

if ABS(A - X • X) < E then X else SQRT(A, (X + A + X) + 2, E).

In an imaginary but recognizable version of LISP {see Levin [8] or

Weissman [9]), this same procedure might be written:

(DEFINE (SQRT AXE)

(IF (LESS (ABS ( - A (. X X))) E) THEN X
E L S E ( S Q R T A ( ÷ ( + X ( + A X ) ) 2 ) E)))

Here, the function names come immediately inside their parentheses.

The clumsiness, for humans, of writing all the parentheses is evident;
the advantages of not having to learn all the conventions, such as
that ( X + A + X ) is ( + X ( ÷ A X ) ) and not ( ÷ ( + X A ) X), is often
overlooked.
It remains to be seen whether a syntax with explicit delimiters is
reactionary, or whether it is the wave of the future. It has important
advantages for editing, interpreting, and for creation of programs by other
programs. The complete syntax of LISP can be learned in an hour or
so; the interpreter is compact and not exceedingly complicated, and
students often can answer questions about the system by reading the
interpreter program itself. Of course, this will not answer all questions
about real, practical implementation, but neither would any feasible
set of syntax rules. Furthermore, despite the language's clumsiness,
many frontier workers consider it to have outstanding expressive power.
Nearly all work on procedures that solve problems by building and

224 MARVIN MINSKY

modifying hypotheses have been written in this or related languages. 1 9 1~ 9

Unfortunately, language designers are generally unfamiliar with this 'I u , i n }~

area, and tend to dismiss it as a specialized body of "symbol- I ,t'¢'l u I't'

manipulation techniques."
Much can be done to clarify the structure of expressions in such
a "syntax-weak" language by using indentation and other layout devices
that are outside the language proper. For example, one can use a
"postponement" symbol that belongs to an input preprocessor to rewrite
the above as

DEFINE (SQRT AXE) ~ .

IF ~ THEN X ELSE ~ .
LESS (ABS ~ ) E.
--A (* XX).
SQRT A ~ E.

+X (+AX)

where the dot means ")(" and the arrow means "insert here the next
expression, delimited by a dot, that is available after replacing (recur-
sively) its own arrows." The indentations are optional. This gets a
good part of the effect of the usual scope indicators and conventions
by two simple devices, both handled trivially by reading programs,
and it is easy to edit because subexpressions are usually complete on
each line.
To appreciate the power and limitations of the postponement
operator, the reader should take his favorite language and his favorite
algorithms and see what happens. He will find many choices of what
to postpone, and he exercises judgment about what to say first, which
arguments to emphasize, and so forth. Of course, t~ is not the answer
to all problems; one needs a postponement device also for list fragments,
and that requires its own delimiter. In any case, these are but steps
toward more graphical program-description systems, for we will not
forever stay confined to mere strings of symbols.
Another expository device, suggested by Dana Scott, is to have
alternative brackets for indicating right-to-left functional composition,
so that one can write (((x)h)g)f instead off(g(h(x))) when one wants
to indicate more naturally what happens to a quantity in the course
of a computation. This allows different "accents," as inf((h(x))g), which
can be read: "Compute f o r what you get by first computing h(x) and
then applying g to it."
The point is better made, perhaps, by analogy than by example. In
their fanatic concern with syntax, language designers have become too
sentence oriented. With such devices as !~ , one can construct objects
that are more like paragraphs, without failing all the way back to
flow diagrams.

Form and Content in Computer Science 225

 Today's high level programming languages offer little expressive
power in the sense of flexibility of style. One cannot control the
sequence of presentation of ideas very much without changing the
algorithm itself.

2.2
Efficiency
and Understanding Programs
What is a compiler for? The usual answers resemble "to translate
from one language to another" or "to take a description of an algorithm
and assemble it into a program, filling in many small details." For the
future, a more ambitious view is required. Most compilers will be
systems that "produce an algorithm, given a description of its effect."
This is already the case for modern picture-format systems; they do
all the creative work, while the user merely supplies examples of the
desired formats: here the compilers are more expert than the users.
Pattern-matching languages are also good examples. But except for
a few such special cases, the compiler designers have made little
progress in getting good programs written. Recognition of common
subexpressions, optimization of inner loops, allocation of multiple
registers, and so forth, lead but to small linear improvements in
efficiency--and compilers do little enough about even these. Automatic
storage assignments can be worth more. But the real payoff is in analysis
of the computational content of the algorithm itself, rather than the way
the programmer wrote it down. Consider, for example:

DEFINE FIB(N): if N = 1 then 1, if N = 2 then 1,

else FIB(N-1) + FIB(N-2).

This recursive definition of the Fibonacci numbers 1, 1, 2, 3, 5, 8,

13," • • can be given to any respectable algorithmic language and will
result in the branching tree of evaluation steps shown in Figure 1.

F(6)

F(5) F(4)

F(4) F(3) F(3) FF(2)

F(3) F(2) F(2) F(1) F(2) F(1)

F(2) F(1)
FIGURE 1

226 MARVINMINSKY
 One sees that the amount of work the machine will do grows I 9 (~ 9
exponentially with N. (More precisely, it passes through the order of 'I llmilig
Aw~Hd
FIB(N) evaluations of the definition!) There are better ways to compute Lt'{'I u l t.
this function. Thus we can define two temporary registers and evaluate
FIB(N1 1) in

DEFINE FIB(NAB): if N= 1 then A else FIB(N- 1A +BA).

which is singly recursive and avoids the branching tree, or even use

A=0
B=I
LOOP SWAP AB
if N = 1 return A
N=N-1
B=A+B
goto LOOP

Any programmer will soon think of these, once he sees what happens
in the branching evaluation. This is a case in which a "course-of values"
recursion can be transformed into a simple iteration. Today's compilers
don't recognize even simple cases of such transformations, although
the reduction in exponential order outweighs any possible gains in local
"optimization" of code. It is no use protesting either that such gains
are rare or that such matters are the programmer's responsibility. If
it is important to save compiling tme, then such abilities could be
excised. For programs written in the pattern-matching languages, for
example, such simplifications are indeed often made. One usually wins
by compiling an efficient tree-parser for BNF system instead of
executing brute force analysis-by-synthesis.
To be sure, a systematic theory of such transformations is difficult.
A system will have to be pretty smart to detect which transformations
are relevant and when it pays to use them. Since the programmer
already knows his intent, the problem would often be easier if the
proposed algorithm is accompanied (or even replaced) by a suitable
goal-declaration expression.
To move in this direction, we need a body of knowledge about
analyzing and synthesizing programs. On the theoretical side there is
now a lot of activity studying the equivalence of algorithms and
schemata, and on proving that procedures have stated properties. On
the practical side the works of W. A. Martin [10] and J. Moses [11]
illustrate how to make systems that know enough about symbolic
transformations of particular mathematical techniques to significantly
supplement the applied mathematical abilities of their users.

Form and Content in Computer Science 227

 There is no practical consequence to the fact that the program-
reduction problem is recursively unsolvable, in general. In any case
one would expect programs eventually to go far beyond human ability
in this activity and make use of a large body of program transforma-
tions in formally purified forms. These will not be easy to apply directly.
Instead, one can expect the development to follow the lines we have
seen in symbolic integration, e.g., as in Slagle [12] and Moses [11].
First a set of simple formal transformations that correspond to the
elementary entries of a Table of Integrals was developed. On top of these
Slagle built a set of heuristic techniques for the algebraic and analytic
transformation of a practical problem into those already understood
elements; this involved a set of characterization and matching pro-
cedures that might be said to use "pattern recognition." In the system
of Moses both the matching procedures and the transformations were
so refined that, in most practical problems, the heuristic search strategy
that played a large part in the performance of Slagle's program became
a minor augmentation of the sure knowledge and its skillful applica-
tion comprised in Moses' system. A heuristic compiler system will
eventually need much more general knowledge and common sense than
did the symbolic integration systems, for its goal is more like making
a whole mathematician than a specialized integrator.

2.3
Describing
Programming Systems
No matter how a language is described, a computer must use a pro-
cedure to interpret it. One should remember that in describing a language
the main goal is to explain how to write programs in it and what such
programs mean. The main goal isn't to describe the syntax.
Within the static framework of syntax rules, normal forms, Post
productions, and other such schemes, one obtains the equivalents of
logical systems with axioms, rules of inference, and theorems. To design
an unambiguous syntax corresponds then to designing a mathematical
system in which each theorem has exactly one proof! But in the
computational framework, this is quite beside the point. One has an
extra ingredient--control--that lies outside the usual framework of a
logical system; an additional set of rules that specify when a rule of
inference is to be used. So, for many purposes, ambiguity is a
pseudoproblem. If we view a program as a process, we can remember
that our most powerful process-describing tools are programs
themselves, and they are inherently unambiguous.
There is no paradox in defining a programming language by a
program. The procedural definition must be understood, of course. One
can achieve this understanding by definitions written in another
language, one that may be different, more familiar, or simpler than
the one being defined. But it is often practical, convenient, and proper

228 MARVIN MINSKY

to use the same language! For to understand the definition, one needs
to know only the working of that particular program, and not all "Iu ri ng
LX~*~ +| |1||
implications of all possible applications of the language. It is this l,et'hnrt.
particularization that makes bootstrapping possible, a point that often
puzzles beginners as well as apparent authorities.
Using BNF to describe the formation of expressions may be retarding
development of new languages that smoothly incorporate quotation,
self-modification, and symbolic manipulation into a traditional
algorithmic framework. This, in turn, retards progress toward problem-
solving, goal-oriented programming systems. Paradoxically, though
modern programming ideas were developed because processes were
hard to depict with classical mathematical notations, designers are turn-
ing back to an earlier form--the equation--in just the kind of situation
that needs program. In Section 3, which is on education, a similar situa-
tion is seen in teaching, with perhaps more serious consequences.

3
Learning, Teaching,
and the "New Mathematics"
Education is another area in which the computer scientist has
confused form and content, but this time the confusion concerns his
professional role. He perceives his principal function to provide pro-
grams and machines for use in old and new educational schemes. Well
and good, but I believe he has a more complex responsibility--to work
out and communicate models of the process of education itself.
In the discussion below, I sketch briefly the viewpoint Ideveloped
with Seymour Papert) from which this belief stems. The following
statements are typical of our view:

-- To help people learn is to help them build, in their heads, various

kinds of computational models.
--This can best be done by a teacher who has, in his head, a
reasonable model of what is in the pupil's head.
-- For the same reason the student, when debugging his own models
and procedures, should have a model of what he is doing, and must
know good debugging techniques, such as how to formulate simple but
critical test cases.
--It will help the student to know something about computational
models and programming. The idea of debugging 2 itself, for example,

ZTuring w a s quite good at d e b u g g i n g h a r d w a r e . He w o u l d leave the p o w e r on, so as not

to lose the "feel" of the thing. Everyone does that today, but it is not t he s a m e thing n o w
that the circuits all w o r k on three or five volts.

Form and Cont e nt in C o m p u t e r Science 229

is a very powerful concept--in contrast to the helplessness promoted
by our cultural heritage about gifts, talents, and aptitudes. The latter
encourages "I'm not good at this" instead of "How can I make myself
better at it?"
These have the sound of common sense, yet they are not among
the basic principles of any of the popular educational schemes such as
"operant reinforcement," "discovery methods," audio-visual synergism,
etc. This is not because educators have ignored the possibility of mental
models, but because they simply had no effective way, before the begin-
ning of work on simulation of thought processes, to describe, construct,
and test such ideas.
We cannot digress here to answer skeptics who feel it too
simpleminded (if not impious, or obscene) to compare minds with
programs. We can refer many such critics to Turing's paper [13]. For
those who feel that the answer cannot lie in any machine, digital
or otherwise, one can argue [14] that machines, when they become
intelligent, very likely will feel the same way. For some overviews of
this area, see Feigenbaum and Feldman [15] and Minsky [16]; one can
keep really up-to-date in this fast-moving field only by reading the
contemporary doctoral theses and conference papers on artificial
intelligence.
There is a fundamental pragmatic point in favor of our propositions.
The child needs models: to understand the city he may use the organism
model; it must eat, breathe, excrete, defend itself, etc. Not a very good
model, but useful enough. The metabolism of a real organism he can
understand, in turn, by comparison with an engine. But to model his
own self he cannot use the engine or the organism or the city or the
telephone switchboard; nothing will serve at all but the computer with
its programs and their bugs. Eventually, programming itself will become
more important even than mathematics in early education. Nevertheless
I have chosen mathematics as the subject of the remainder of this paper,
partly because we understand it better but mainly because the prejudice
against programming as an academic subject would provoke too much
resistance. Any other subject could also do, I suppose, but mathematical
issues and concepts are the sharpest and least confused by highly
charged emotional problems.

3.1
Mathematical Portrait
of a Small Child
Imagine a small child of between five and six years, about to enter
the first grade. If we extrapolate today's trend, his mathematical educa-
tion will be conducted by poorly oriented teachers and, partly, by poorly
programmed machines; neither will be able to respond to much beyond
"correct" and "wrong" answers, let alone to make reasonable inter-

230 MARVIN MINSKY

pretations of what the child does or says, because neither will contain
good models of the children, or good theories of children's intellectual
development. The child will begin with simple arithmetic, set theory,
and a little geometry; ten years later he will know a little about the
formal theory of the real numbers, a little about linear equations, a
little more about geometry, and almost nothing about continuous and
limiting processes. He will be an adolescent with little taste for analytical
thinking, unable to apply the ten years' experience to understanding
his new world.
Let us look more closely at our young child, in a composite picture
drawn from the work of Piaget and other observers of the child's mental
construction.
Our child will be able to say "one, two, three . . . . " at least up
to thirty and probably up to a thousand. He will know the names of
some larger n u m b e r s but will not be able to see, for example, w h y
ten thousand is a h u n d r e d hundred. He will have serious difficulty
in counting backwards unless he has recently b e c o m e very interested
in this. {Being good at it would make simple subtraction easier, and
might be worth some practice.) He doesn't have m u c h feeling for odd
and even.
He can count four to six objects with perfect reliability, but he will
not get the same count every time with fifteen scattered objects. He
will be annoyed with this, because he is quite sure he should get the
same n u m b e r each time. The observer will therefore think the child
has a good idea of the n u m b e r concept but that he is not too skillful
at applying it.
However, important aspects of his concept of n u m b e r will not be
at all secure by adult standards. For example, w h e n the objects are
rearranged before his eyes, his impression of their quantity will be
affected by the geometric arrangement. Thus he will say that there are
fewer x's than y's in:

XXXXXXX

YYYYYYY

and w h e n we move the x's to

X X X X X X X

YYYYYYY

he will say there are more x's than y's. To be sure, he is answering [in
his own mind) a different question about size, quite correctly, but this
is exactly the point; the immutability of the number, in such situations,
has little grip on him. He cannot use it effectively for reasoning although
he shows, on questioning, that he knows that the n u m b e r of things
cannot change simply because they are rearranged. Similarly, w h e n

Form and Content in C o m p u t e r Science 231

water is p o u r e d from one glass to another (Figure 2(a)), he will say
that there is more water in the tall jar than in the squat one. He will
have poor estimates about plane areas, so that we will not be able to
find a context in which he treats the larger area in Figure 2 (b) as four
times the size of the smaller one.

(a)

£ A
(b) (c)
FIGURE 2

W h e n he is an adult, by the way, and is given two vessels, one twice

as large as the other, in all dimensions (Figure 2 (c)), he will think the
one holds about four times as m u c h as the other; probably he will never
acquire better estimates of volume.
As for the numbers themselves, we know little of what is in his mind.
According to Galton [17], thirty children in a h u n d r e d will associate
small n u m b e r s with definite visual locations in the space in front of
their body image, arranged in some idiosyncratic m a n n e r such as that
shown in Figure 3. T h e y will probably still retain these as adults, and
m a y use t h e m in some obscure semiconscious way to r e m e m b e r
telephone numbers; they will probably grow different spatio-visual
representations for historical dates, etc. The teachers will never have

FIGURE 3

232 MARVIN MINSKY

heard of such a thing and, if a child speaks of it, even the teacher with 1 9 I; 9
her own n u m b e r form is unlikely to respond with recognition. {My 'luring
/~%%r;II (I
experience is that it takes a series of carefully posed questions before I+t't'lu rt"
one of these adults will respond, "Oh, yes; 3 is over there, a little farther
back.") W h e n our child learns column sums, he m a y keep track of
carries by setting his tongue to certain teeth, or use some other obscure
device for temporary memory, and no one will ever know. Perhaps some
ways are better than others.
His geometric world is different from ours. He does not see clearly
that triangles are rigid, and thus different from other polygons. He does
not k n o w that a 100-line approximation to a circle is indistinguishable
from a circle unless it is quite large. He does not draw a cube in perspec-
tive. He has only recently realized that squares become diamonds w h e n
put on their points. The perceptual distinction persists in adults. Thus
in Figure 4 we see, as noted by Attneave [18], that the impression of
square versus diamond is affected by other alignments in the scene,
evidently by determining our choice of which axis of s y m m e t r y is to
be used in the subjective description.

FIGURE 4

Our child understands the topological idea of enclosure quite well.

Why? This is a v e r y complicated concept in classical mathematics but
in terms of computational processes it is perhaps not so difficult. But
our child is almost sure to be m u d d l e d about the situation in Figure 5
(see Papert [19]): " W h e n the bus begins its trip around the lake, a boy

FIGURE 5

Form and Content in Computer Science 233

is seated on the side away from the water. Will he be on the lake side
at some time in the trip?" Difficulty with this is liable to persist through
the child's eighth year, and perhaps relates to his difficulties with other
abstract double reversals such as in subtracting negative numbers, or
with apprehending other consequences of continuity--'Nt what point
in the trip is there any sudden change?"--or with other bridges between
logical and global.
Our portrait is drawn in more detail in the literature on develop-
mental psychology. But no one has yet built enough of a computational
model of a child to see how these abilities and limitations link together
in a structure compatible with land perhaps consequential to I other
things he can do so effectively. Such work is beginning, however, and
I expect the next decade to see substantial progress on such models.
If we knew more about these matters, we might be able to help the
child. At present we don't even have good diagnostics: his apparent
ability to learn to give correct answers to formal questions may show
only that he has developed some isolated library routines. If these
cannot be called by his central problem-solving programs, because they
use incompatible data structures or whatever, we may get a high rated
test-passer who will never think very well.
Before computation, the community of ideas about the nature of
thought was too feeble to support an effective theory of learning and
development. Neither the finite-state models of the Behaviorists, the
hydraulic and economic analogies of the Freudians, nor the rabbit-in-
the-hat insights of the Gestaltists supplied enough ingredients to under-
stand so intricate a subject. It needs a substrate of already debugged
theories and solutions of related but simpler problems. Now we have
a flood of such ideas, well defined and implemented, for thinking about
thinking; only a fraction are represented in traditional psychology:
symbol table closed subroutines
pure procedure pushdown list
time-sharing interrupt
calling sequence communication cell
functional argument common storage
memory protection decision tree
dispatch table hardware-software trade-off
error message serial-parallel trade-off
function-call trace time-memory trade-off
breakpoint conditional breakpoint
languages asynchronous processor
compiler interpreter
indirect address garbage collection
macro list structure
property list block structure
data type look-ahead
hash coding look-behind
microprogram diagnostic program
format matching executive program

234 MARVINMINSKY
 These are just a few ideas from general systems programming and I 9 (i 9
debugging; we have said nothing about the many more specifically 'i Inn'|rig
Awi,,'d
relevant concepts in languages or in artificial intelligence or in computer I,e¢'ture
hardware or other advanced areas. All these serve today as tools
of a curious and intricate craft, programming. But just as astronomy
succeeded astrology, following Kepler's regularities, the discovery
of principles in empirical explorations of intellectual process in
machines should lead to a science. (In education we face still the same
competition! The Boston Globe has an astrology page in its "comics"
section. Help fight intellect pollution!)
To return to our child, how can our computational ideas help him
with his number concept? As a baby he learned to recognize certain
special pair configurations such as two hands or two shoes. Much later
he learned about some threes-- perhaps the long gap is because the
environment doesn't have many fixed triplets: if he happens to find three
pennies he will likely lose or gain one soon. Eventually he will find
some procedure that manages five or six things, and he will be less at
the mercy of finding and losing. But for more than six or seven things,
he will remain at the mercy of forgetting; even if his verbal count is
flawless, his enumeration procedure will have defects. He will skip some
items and count others twice. We can help by proposing better procedures;
putting things into a box is nearly foolproof, and so is crossing them
off. But for fixed objects he will need some mental grouping procedure.
First one should try to know what the child is doing; eye-motion
study might help, asking him might be enough. He may be selecting the
next item with some unreliable, nearly random method, with no good
way to keep track of what has been counted. We might suggest: sliding
a cursor; inventing easily remembered groups; drawing a coarse mesh.
In each case the construction can be either real or imaginary. In using
the mesh method one has to remember not to count twice objects that
cross the mesh lines. The teacher should show that it is good to plan
ahead, as in Figure 6, distorting the mesh to avoid the ambiguities!

• p • ~o •1 /o

Oi •
• • • • • • •
P • • i
FIGURE 6

Mathematically the important concept is that "every proper counting

procedure yields the same number." The child will understand that any
algorithm is proper which (1) counts all the objects, (2) counts none of
them twice.

Form and Content in Computer Science 235

 Perhaps this procedural condition seems too simple; even an adult
could understand it. In any case, it is not the concept of number adopted
in what is today generally called the "New Math," and taught in our
primary schools. The following polemic discusses this.

3.2
The "New M a t h e m a t i c s "
By the "new math" I mean certain primary school attempts to
imitate the formalistic outputs of professional mathematicians.
Precipitously adopted by many schools in the wake of broad new
concerns with early education, I think the approach is generally bad
because of form-content displacements of several kinds. These cause
problems for the teacher as well as for the child.
Because of the formalistic approach the teacher will not be able to
help the child very much with problems of formulation. For she will
feel insecure herself as she drills him on such matters as the difference
between the empty set and nothing, or the distinction between the
"numeral" 3 + 5 and the numeral 8 which is the "common name" of
the number eight, hoping that he will not ask what is the common name
of the fraction 8/1, which is probably different from the rational 8/1
and different from the quotient a/1 and different from the "indicated
division" 8/1 and different from the ordered pair {8,1). She will be
reticent about discussing parallel lines. For parallel lines do not usually
meet, she knows, but they might (she has heard) if produced far enough,
for did not something like that happen once in an experiment by some
Russian mathematicians? But enough of the problems of the teacher:
let us consider now three classes of objections from the child's stand-
point.

D e v e l o p m e n t a l Objections. It is very bad to insist that the child

keep his knowledge in a simple ordered hierarchy. In order to retrieve
what he needs, he must have a multiply connected network, so that
he can try several ways to do each thing. He may not manage to match
the first method to the needs of the problem. Emphasis on the "formal
proof" is destructive at this stage, because the knowledge needed for
finding proofs, and for understanding them, is far more complex (and
less useful) than the knowledge mentioned in proofs. The network of
knowledge one needs for understanding geometry is a web of examples
and phenomena, and observations about the similarities and differences
between them. One does not find evidence, in children, that such webs
are ordered like the axioms and theorems of a logistic system, or that
the child could use such a lattice if he had one. After one understands
a phenomenon, it may be of great value to make a formal system for
it, to make it easier to understand more advanced things. But even
then, such a formal system is just one of many possible models; the
New Math writers seem to confuse their axiom-theorem model with

236 MARVIN MINSKY

the n u m b e r system itself. In the case of the axioms for arithmetic, I I ~) (~ I)

will now argue, the formalism is often likely to do more harm than good " l uri ng
for the understanding of more advanced things. I,c('lmc
Historically, the "set" approach used in New Math comes from a
formalist attempt to derive the intuitive properties of the c o n t i n u u m
from a nearly finite set theory. They partly succeeded in this stunt tor
"hack," as some programmers would put it), but in a m a n n e r so complex
that one cannot talk seriously about the real n u m b e r s until well into
high school, if one follows this model. The ideas of topology are deferred
until m u c h later. But children in their sixth year already have well-
developed geometric and topological ideas, only they have little ability
to manipulate abstract symbols and definitions. We should build out
from the child's strong points, instead of undermining him by
attempting to replace what he has by structures he cannot yet handle.
But it is just like m a t h e m a t i c i a n s - - c e r t a i n l y the world's worst
e x p o s i t o r s - - t o think: "You can teach a child anything, if you just get
the definitions precise enough," or "If we get all the definitions right
the first time, we won't have any trouble later." We are not program-
ming an e m p t y machine in FORTRAN: we are meddling with a poorly
understood large system that, characteristically, uses multiply defined
symbols in its normal heuristic behavior.

I n t u i t i v e O b j e c t i o n s . New Math emphasizes the idea that a num-

ber can be identified with an equivalence class of all sets that can be
put into one-to-one correspondence with one another. Then the rational
n u m b e r s are defined as equivalence classes of pairs of integers, and
a maze of formalism is introduced to prevent the child from identi~ing
the rationals with the quotients or fractions. Functions are often treated
as sets, although some texts present "function machines" with a super-
ficially algorithm{c flavor. The definition of a "variable" is another
fiendish maze of complication involving names, values, expressions,
clauses, sentences, numerals, "indicated operations," and so forth.
(In fact, there are so many different kinds of data in real problem-solving
that real-life mathematicians do not usually give t h e m formal distinc-
tions, but use the entire problem context to explain them.) In the course
of pursuing this formalistic obsession, the curriculum never presents
any coherent picture of real mathematical p h e n o m e n a of processes,
discrete or continuous; of the algebra whose notational syntax concerns
it so; or of geometry. The " t h e o r e m s " that are "proved" from time to
time, such as, "A n u m b e r x has only one additive inverse, - x , " are so
m u n d a n e and obvious that neither teacher nor student can make out
the purpose of the proof. The "official" proof would add y to both sides
of x + ( - y) = 0, apply the associative law, then the commutative law,
then the y = ( - y) = 0 law, and finally the axioms of equality, to show
that y must equal x. The child's mind can more easily understand
deeper ideas: "In x + ( - y ) = 0, if y were less than x there would be

Form and C ont e nt in C o m p u t e r Science 237

some left over; while if x were less than y there would be a minus
number left-- so they must be exactly equal:' The child is not permitted
to use this kind of order-plus-continuity thinking, presumably because
it uses "more advanced knowledge," hence isn't part of a "real proof."
But in the network of ideas the child needs, this link has equal logical
status and surely greater heuristic value. For another example, the
student is made to distinguish clearly between the inverse of addition
and the opposite sense of distance, a discrimination that seems entirely
against the fusion of these notions that would seem desirable.
C o m p u t a t i o n a l Objections. The idea of a procedure, and the
know-how that comes from learning how to test, modify, and adapt
procedures, can transfer to many of the child's other activities. Tradi-
tional academic subjects such as algebra and arithmetic have relatively
small developmental significance, especially when they are weak in in-
tuitive geometry. (The question of which kinds of learning can
"transfer" to other activities is a fundamental one in educational theory:
I emphasize again our conjecture that the ideas of procedures and
debugging will turn out to be unique in their transferability.) In algebra,
as we have noted, the concept of "variable" is complicated; but in
computation the child can easily see "x + y + z" as describing a pro-
cedure (any procedure for adding!) with "x," "y," and "z" as pointing
to its "data:' Functions are easy to grasp as procedures, hard if imagined
as ordered pairs. If you want a graph, describe a machine that draws
the graph; if you have a graph, describe a machine that can read it to
find the values of the function. Both are easy and useful concepts.
Let us not fall into a cultural trap; the set theory "foundation" for
mathematics is popular today among mathematicians because it is the
one they tackled and mastered lin college). These scientists simply are
not acquainted, generally, with computation or with the Post-Turing-
McCulloch-Pitts-McCarthy-Newell-Simon-... family of theories that
will be so much more important when the children grow up. Set theory
is not, as the logicians and publishers would have it, the only and true
foundation of mathematics; it is a viewpoint that is pretty good for
investigating the transfinite, but undistinguished for comprehending
the real numbers, and quite substandard for learning about arithmetic,
algebra, and geometry.
To summarize my objections, the New Math emphasized the use
of formalism and symbolic manipulation instead of the heuristic and
intuitive content of the subject matter. The child is expected to learn
how to solve problems but we do not teach him what we know, either
about the subject or about problem-solving, a
3In a shrewd but hilarious discussion of New Math textbooks, Feynman [20] explores
the consequences of distinguishing between the thing and itself. "Color the picture of
the ball red," a book says, instead of "Color the ball red." "Shall we color the entire square
area in which the ball image appears or just the part inside the circle of the ball?" asks
Fegnman. ITo "color the balls red" would presumably have to be "color the insides of
the circles of all the members of the set of balls" or something like that. I

238 MARVIN MINSKY

 As an e x a m p l e of h o w the p r e o c c u p a t i o n w i t h f o r m (in this case, 1 9 6 9
the a x i o m s for arithmetic) can w a r p o n e ' s v i e w of the content, let us 'luring
Aw~nd
e x a m i n e the w e i r d c o m p u l s i o n to insist that addition is u l t i m a t e l y an I,r¢lurc
operation on just t w o quantities. In N e w Math, a + b + c m u s t "really"
be one of ( a + ( b + c ) ) or ( ( a + b ) + c ) , a n d a + b + c + d can be m e a n -
ingful o n l y after several applications of the associative law. N o w this
is silly in m a n y contexts• T h e child has a l r e a d y a g o o d intuitive idea
of w h a t it m e a n s to p u t several sets together; it is just as e a s y to m i x
five colors of b e a d s as two. T h u s addition is a l r e a d y an n-ary operation•
But listen to the b o o k trying to p r o v e that this is not so:

Addition is . . . always performed on two numbers. This may not seem

reasonable at first sight, since you have often added long strings of figures. Try
an experiment on yourself. Try to add the numbers 7, 8, 3 simultaneously. No
matter how you attempt it, you are forced to choose two of the numbers, add
them, and then add the third to their sum.
--From a ninth-grade text

Is the height of a t o w e r the result of a d d i n g its stages b y pairs in

a certain order? Is the length or area of an object p r o d u c e d that w a y
f r o m its parts? W h y did t h e y i n t r o d u c e their sets a n d their o n e - o n e
c o r r e s p o n d e n c e s t h e n to miss the point? Evidently, t h e y have talked
themselves into believing that the axioms t h e y selected for algebra have
s o m e special k i n d of truth!
Let us c o n s i d e r a f e w i m p o r t a n t a n d p r e t t y ideas that are not
discussed m u c h in grade school. First c o n s i d e r the s u m 1/z + 1/4 + 1/8 +
• • •. I n t e r p r e t e d as area, one gets fascinating r e g r o u p i n g ideas, as in
Figure 7. O n c e the child k n o w s h o w to do division, he c a n c o m p u t e

FIGURE 7

a n d appreciate s o m e quantitative aspects of the limiting p r o c e s s .5, .75,

.875, .9375, .96875, • • ', a n d he can learn a b o u t folding a n d cutting
a n d e p i d e m i c s a n d populations. He c o u l d learn a b o u t x = p x + qx,
w h e r e p + q = 1, a n d h e n c e appreciate dilution; he can learn that
3/4, 4/5, s/6, 6/7, 7/8, • • • -~ 1 a n d begin to u n d e r s t a n d the m a n y colorful
a n d c o m m o n - s e n s e g e o m e t r i c a l a n d topological c o n s e q u e n c e s of s u c h
matters•
But in the N e w Math, the syntactic distinctions b e t w e e n rationals,
quotients, a n d fractions are carried so far that to see w h i c h of 3/8 a n d
4/9 is larger, one is not p e r m i t t e d to c o m p u t e a n d c o m p a r e .375 w i t h

Form and Content in Computer Science 239

 .4444. O n e m u s t cross-multiply. N o w cross-multiplication is v e r y cute,
but it has two bugs: (1) no one can r e m e m b e r w h i c h w a y the resulting
conditional should branch, and (2) it doesn't tell h o w far apart the
n u m b e r s are. The abstract concept of order is v e r y elegant (another set
of axioms for the obvious) but the children already u n d e r s t a n d order
pretty well and w a n t to k n o w the amounts.
Another obsession is the concern for n u m b e r base. It is good for the
children to understand clearly that 223 is "two h u n d r e d " plus "twenty"
plus "three," and I think that this should be m a d e as simple as possible
rather t h a n complicated.4 I do not think that the idea is so rich that
one should drill young children to do arithmetic in several bases! For
there is v e r y little transfer of this feeble concept to other things, and
it risks a crippling insult to the fragile arithmetic of pupils who, already
troubled with 6 + 7 = 13, n o w find that 6 + 7 = 15. Besides, for all
the attention to n u m b e r base, I do not see in m y children's books any
concern with even a few nontrivial i m p l i c a t i o n s - - c o n c e p t s that might
justify the attention, such as:

Why is there only one way to write a decimal integer?

Why does casting out nines work? (It isn't even mentioned.)
What happens if we use arbitrary nonpowers, such as a + 37b + 24c + 1ld +...
instead of the usual a + 10b + 100c + 1000d +... ?

If they don't discuss such matters, they m u s t have a n o t h e r purpose.

M y conjecture is that the whole fuss is to m a k e the kids better under-
stand the p r o c e d u r e s for multiplying and dividing. But f r o m a develop-
m e n t a l v i e w p o i n t this m a y be a serious m i s t a k e - - i n the strategies
of both the old and the " n e w " m a t h e m a t i c a l curricula. At best, the
standard algorithm for long division is c u m b e r s o m e , and most children
will n e v e r use it to explore n u m e r i c p h e n o m e n a . And, although it is
of s o m e interest to u n d e r s t a n d h o w it works, writing out the w h o l e
display suggests that the e d u c a t o r believes that the child ought to
u n d e r s t a n d the horrible thing e v e r y time! This is wrong. The impor-
tant idea, if any, is the r e p e a t e d subtraction; the rest is just a clever
but not vital p r o g r a m m i n g hack.
If we can teach, p e r h a p s by rote, a practical division algorithm,
fine. But in a n y case let us give t h e m little calculators; if that is too
expensive, w h y not slide rules. Please, w i t h o u t an impossible explana-
tion. The i m p o r t a n t thing is to get on to the real n u m b e r s ! The N e w
Math's concern with integers is so fanatical that it r e m i n d s me, if I m a y
mention another pseudoscience, of numerology. (How about that, Boston
Globe!)
The Cauchy-Dedekind-Russell-Whitehead set-theory formalism was
a large a c c o m p l i s h m e n t - - a n o t h e r (following Euclid) of a series of
d e m o n s t r a t i o n s that m a n y m a t h e m a t i c a l ideas can be derived f r o m a
few primitives, albeit by a long and tortuous route. But the child's
4Cf. Tom Lehrer's song, "New Math" [21].

240 MARVIN MINSKY

p r o b l e m is to acquire the ideas at all; he needs to learn about reality.
In terms of the concepts available to him, the entire formalism of set '1 ,aninlg
/~,l,v;m mI|
t h e o r y cannot hold a candle to one older, simpler, and possibly greater
l A'¢'I II n'l"
idea: the nonterminating decimal representation of the intuitive real
n u m b e r line.
There is a real conflict between the logician's goal and the educator's.
The logician wants to minimize the variety of ideas, and doesn't mind a long,
thin path. The educator {rightly} wants to make the paths short and doesn't
mind--in fact, prefers--connections to many other ideas. A n d he cares
almost not at all about the directions of the links.
As for better understanding of the integers, countless exercises
in making little children draw diagrams of one-one correspondences
will not help, I think. It will help, no doubt, in their learning valuable
algorithms, not for n u m b e r but for the important topological and pro-
cedural problems in drawing paths without crossing, and so forth. It
is just that sort of problem, n o w treated entirely accidentally, that we
should attend to.
The c o m p u t e r scientist thus has a responsibility to education. Not,
as he thinks, because he will have to program the teaching machines.
Certainly not because he is a skilled user of "finite mathematics." He
knows h o w to d e b u g programs; he must tell the educators how to help
the children to debug their o w n problem-solving processes. He knows
how procedures depend on their data structures; he can tell educators
h o w to prepare children for n e w ideas. He knows w h y it is bad to use
double-purpose tricks that h a u n t one later in debugging and enlarging
programs. {Thus, one can capture the kids' interest by associating small
numbers with arbitrary colors. But what will this trick do for their later
attempts to apply n u m b e r ideas to area, or to volume, or to value?) The
c o m p u t e r scientist is the one who must study such matters, because
he is the proprietor of the concept of procedure, the secret educators
have so long been seeking.

References
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struction du Nombre, Presses Universitaires de France, Paris, 1960.
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Warner Bros. Records.
Categories and Subject Descriptors:
D.3.1 [Software]: Formal Definitions and Theory--syntax; D.3.4
[Software]: Processors--compilers; F.2.1 [Theory of Computation]:
Numerical Algorithms and Problems--computations on matrices;
F.4.1 [Theory of Computation]: Mathematical Logic -- recursive function
theory; 1.2.6 [Computing Methodologies]: Learning--concept learning;
K.3.0 [Computing Milieux]: Computers and Education--general
General Terms:
Algorithms, Languages, Theory
Key Words and Phrases:
Heuristic programming, new math

242 MARVIN MINSKY