COMPUTERS AND MATHEMATICAL NOTATION
Kenneth E. Iverson
By relieving the brain of all unnecesary work, a good notation
sets it free to concentrate on more advanced problems, and in
effect increases the mental power of the race.
A. N. Whitehead
Some
symbols, like that were used originally
for only positive integral values of n stimulated intellectual
experimentation when n is fractional, negative, or complex,
which led to vital extensions of ideas.
F. Cajori
Mathematical
notation, like language, has grown up without much looking to,
at the dictates of convenience and with the sanction of the majority.
A. de Morgan
A. INTRODUCTION
The foregoing quotations assert both the power
and the muddle of mathematical notation. They may be found in
the concluding section of Volume II of Cajori, A History of
Mathematical Notations [1]. Cajori also says: "In
the light of the teaching of history it is clear that new forces
must be brought into action in order to safeguard the future against
the play of blind chance. The drift and muddle of the past is
intolerable. We believe that this new agency will be organization
and co-operation."
Writing in the 1920s, Cajori could not
have foreseen the potential impact of the computer, and of the
programming languages that have developed to apply it to mathematical
problems.
For practical reasons, programming languages have largely
adopted a standard alphabet (or character set, known as
ASCII), unambiguous use of symbols (within any given language),
and the discipline of a precisely defined grammar, or syntax.
In spite of the wide use of programming languages in expressing
and executing mathematical problems, the standardization and other
notions from programming languages have had virtually no impact
on the mathematical notation used in the exposition and teaching
of mathematics.
It might be argued that mathematical notation (to
be referred to as MN) is adequate as it is, and could not
benefit from the infusion of ideas from programming languages.
However, MN suffers an important defect: it is not executable
on a computer, and cannot be used for rapid and accurate exploration
of mathematical notions.
The remainder of this paper is a critique
of MN, together with suggestions for improvement based
on experience in the design of programming languages and in their
use in the exposition of topics in mathematics.
In any criticism,
it is important that suggestions for improvement come from a coherent
and tested system, lest they be dismissed as disconnected, and
possibly conflicting, proposals. For this reason I will base all
suggestions on a single programming system. The system J
is chosen for the following reasons:
- It is available free from website http://www.jsoftware.com
- The printed documentation [2-5] is also available on-line.
It can be displayed by clicking on the HELP menu, and selections
from it can be printed for perusal. In particular, the one-page
Vocabulary provides summary definitions of the entire language. - It has been used in the exposition of topics ranging from
arithmetic to calculus to fractals [6 7], and most of these texts
are also available on-line. - All expressions in the paper are executable, and the serious
reader can experiment with them, with any component phrases, and
with any variants that may suggest themselves.
The difficulty posed
for the reader by the use of J (or any programming language)
is that of understanding an unfamiliar notation. This will be
eased by the fact that each expression can be executed, that its
results are shown, and that most are accompanied by comments (in
a distinct font).
We will follow the common convention of using
italics in the names in MN, and a uniformly spaced font
for expressions in programming languages. Any expression to be
entered is shown indented, and its result is not indented.
B. RELATIONS AND ASSIGNMENT
Mathematics makes much use of relations
such as less than (denoted by <), equals
(=), and greater than (>); and the result
of a relation is interpreted as true if the relation holds,
and false otherwise.
Boole [8] denoted true by 1
and false by 0, and (by analogy with the behaviour
of multiplication on these arguments) used the symbol for multiplication
to denote logical and. By a looser analogy, he used +
for logical or.
Later workers, wishing to use logical functions
together with arithmetic functions, introduced distinct symbols:
(from the Latin vel) for
or, and for and.
For the same reason, J (being confined to the ASCII alphabet)
uses +. for or, and *. for and. For
example:
0 +. 0 The proposition p or q is false if both p and q are false
0
0 +. 1 The proposition p or q is true if p is false and q is true
1
1 +. 0
1
1 +. 1
1
1 *. 1 p and q is true if both p and q are true
1
MN
uses the symbol = for a relation, but also uses it for
assignment, as in the expression (Let) x=3. Again,
to denote these two distinct notions without ambiguity, programming
languages use distinct notation (that usually includes the symbol
=), as in := (in ALGOL), and =: (in J).
For example:
a=: 4 a is four
v=: 2 3 5 7 v is the vector of the first four primes
plus=: + plus is the function +
a * v a times v
8 12 20 28
v plus v v plus v
4 6 10 14
C. EXPLORATION
The use of executable notation for exploration
is illustrated by the following passage from Chapter 2 of Exploring
Math [6]:
It is commonly thought that math is about
numbers. So it is, but numbers are not the only, nor even
the most important, concern of math. It would be more accurate
to say that math is concerned with relations,
and with proofs of relations.
Although the
first chapter dealt only with numbers, it should be clear that
the interesting aspects were the relations
between results. For example:
a=: i. 6 The first six non-negative integers
b=: ?.~ 6 The integers in random order
b
5 1 2 4 3 0
3*a
0 3 6 9 12 15
a+a+a
0 3 6 9 12 15
(3*a) = (a+a+a) The relation between multiplication and addition
1 1 1 1 1 1
a = b The lists a and b are not equal
0 1 1 0 0 0
sort=: /:~
sort b
0 1 2 3 4 5
sort a
0 1 2 3 4 5
(sort a)=(sort b) But are similar; one is a permutation of the other
1 1 1 1 1 1
We
will further illustrate this matter of relations
by examples that do not concern numbers. For example, the word
'POST' is said to be an anagram of
the word 'SPOT' because the letters of 'SPOT' can
be permuted to give the word 'POST'. Thus
'SPOT' and 'POST' are similar
in the sense already defined for lists. The similarity of these
words may be tested as follows:
w=: 'SPOT' x=: 'POST' sort w OPST sort x OPST (sort w)=(sort x) 1 1 1 1
Sorting w
produces OPST. Is it an anagram? We
will say that it is, although it is not an English
word.
You could (and should) attempt to write down all distinct
anagrams of 'SPOT', finding a surprising number of English
words among them. However, this might be rather difficult to do;
in a long list of words it is easy to overlook
repetitions, and you may not even know how many anagrams
to expect all together.
We will now use the anagram
function A. for this purpose. Its
left argument chooses one of many permutations
to apply to the list right argument. Thus:
w
SPOT
8 A. w
POST
12 A. 8 A. w The permutation 12 A. is the inverse of 8 A.
SPOT
0 1 2 3 4 5 6 7 8 A. w
SPOT
SPTO
SOPT
SOTP
STPO
STOP
PSOT
PSTO
POST
30 A. w
|index error
| 30 A.w
The last result
shows that there is a limit to the valid left argument; properly
so, since there is a limit to the number of different permutations
of a list. But how many are there? In the case
of a two-item list 'AB' there are clearly only two possibilities,
the identity permutation
that leaves the list unchanged, and the one that gives 'BA'.
Thus:
0 1 A. 'AB' AB BA
Write down all permutations
of the list 'ABC' to convince yourself
that there are six possible permutations. Thus:
(i. 6) A. 'ABC' ABC ACB BAC BCA CAB CBA
D. NUMBER SYSTEMS AND OPERATORS
In discussing number systems
it is common to use a phrase such as " is 555" to
indicate that the base-8 representation of the number of days
in a year is 555. This is predicated on the fact that the 365
is "obviously" expressed in decimal. How, then, do we
indicate the inverse case? what would the result of be?
And what is one to make of the expression " is 365"?
These
matters may be clarified by the use of lists; a number is represented
by a list of digits. For example:
d=: 3 6 5
w=: 100 10 1 Weights for decimal (base 10)
d * w
300 60 5
It
remains to apply addition over the list d*w, a matter
expressed in MN by . The applications of other functions
over a list are denoted similarly, using
for product, for and,
and for or. The general notion is simply the modification
of a function by an operator in the sense introduced by
Heaviside [9].
As used by Heaviside in the treatment of differential
equations, the notion seems too difficult for use at an elementary
level. However, the obvious analogy between a function
in math, and a verb in English, leads to the notion of
a Heaviside operator as the analog of an adverb, something
that modifies a verb to produce a different, though related, verb.
Not only is the term adverb more suggestive than operator,
but it avoids conflict with its use in MN as a synonym
for function. Finally:
+/d*w Sum over product of d and w. The / is an adverb that modifies +
365
The parallel
with the adverb is apropos: in English (and other natural languages),
the use of m verbs together with n adverbs permits the prescription
of m times n related verbs. The adverb is equally prolific in
a mathematical context. For example:
+/ d Sum over ()
*/ d Product over ()
*./ d < w And over ()
+./ d < w Or over ()
>./ d Max (greater of) over
<./ d Min (lesser of) over
In English, a
conjunction (such as the copulative conjunction and in
the phrase run and hide) can apply to two verbs to produce
a related verb. Analogously, an operator in math may apply to
two functions (as in the composition , and in convolution),
or to a verb and a noun, as in Currying or bonding.
For example:
pof=: */ @ ! Product over atop factorial
pof d=: 3 6 5 (!3)*(!6)*(!5)
518400
cube=: ^&3 Power with 3
cube d
27 216 125
^. d Natural logarithm
1.09861 1.79176 1.60944
10&^. d Base 10 logarithm
0.4771213 0.7781513 0.69897
Conjunctions, like adverbs, are very prolific,
providing simple expressions for a host of special notations in
MN. For example, falling and rising factorial
functions (denoted in Concrete Mathematics [11] by underscored
and overscored superscripts) may be expressed in the form of the
product x(x+s)(x+2s)(x+3s)
for s=-1 and s=1, respectively. Since the case s=0
is equivalent to the power function, the factorial functions are
denoted by ^!._1 and ^!.1, using the conjunction
!. (called fit). For example:
x=: 5
m=: 4
x ^!._1 m Falling factorial
120
x*(x-1)*(x-2)*(x-3)
120
*/x,(x-1),(x-2),(x-3)
120
x - i. m
5 4 3 2
*/ x - i. m
120
x ^!.0 m Power
625
rf=: ^!.1 Rising factorial
x rf m
1680
Named for Haskell Curry, who introduced
the notion in discussing Combinators [10].
E. AMBIVALENCE
In chemistry, the number of atoms that will
bind with a given atom is called its valence. Adopting
this notion, we will say that a function such as addition (that
applies to two arguments) has valence 2, that the sum function
+/ derived from it has valence 1, and that (because it
may be used with either one argument or two), the function denoted
by - is ambivalent. The effective valence
of - is determined by context. For example:
d=: 3 6 5 5-d 2 _1 0 -d _3 _6 _5
Not only does ambivalence obviate
a further symbol for negation, but it provides a useful mnemonic:
-d is equivalent to 0-d.
This ambivalence of -
goes unremarked in MN. Moreover, the notion remains unexploited;
it could be used to good effect for a wide class of dyadic functions:
3 % 5 Division
0.6
% 5 Reciprocal (1%5)
0.2
3 ^ 5 Power
243
^ 5 Exponential (e^5, where e is Euler’s number 2.71828…)
148.413
3 ! 5 Binomial coefficient ((!5) % (!3) * (!5-3))
10
!5 Factorial
120
3 < d
0 1 1
< d
3 > d
0 0 0
> < d
3 6 5
The final examples illustrate how
the mnemonic may be visual rather than relational: <
packs its argument into a box, and > opens it.
In
J, all functions are ambivalent, including not only primitive
functions but those derived from them by the use of operators.
For example:
*/ d Times over
90
1 2 3 4 */ 1 2 3 4 Times table
1 2 3 4
2 4 6 8
3 6 9 12
4 8 12 16
pt=: 0 1 2 3 4 !/ 0 1 2 3 4
pt Pascal’s Triangle (with meaningful zeros)
1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1
X=: +/ . * Matrix product produced by the dot operator
pt X pt
1 2 4 8 16
0 1 4 12 32
0 0 1 6 24
0 0 0 1 8
0 0 0 0 1
X pt The monadic case is the permanent
1 (-/ . * is the determinant)
Altogether, ambivalence provides an enormous, unobtrusive,
and mnemonic economy of symbols.
F. MISSING SYMBOLS
In MN, a product may be written in
the form (or, perhaps x*y), but can also
be written as xy or x y, omitting the symbol for
multiplication. The brevity achieved by such elision is attractive,
and is widely used.
Other function symbols are also elided, as
in for power (the function being indicated only by the
raised position of the exponent n), and in MN or
M N for the matrix product (distinguished from the elided
product xy only by the capital letters that must be used
to denote the arguments).
The consequences of such elision are often overlooked:
- Expressions of the form a +/ b and a */ b used
for addition and multiplication tables in J could not be
used for any functions (such as power in MN) not provided
with a symbol. It may be noted that the power table yields the
important Vandermonde matrix ( as in 0 1 2 3 ^/ 0 1 2 3). - Multi-character names (as in bob=: 7 and pat=: 5
for the ages of two children, and bob-pat for the difference)
cannot be used because they would denote products. This may explain
why such mnemonic names are not used in teaching elementary algebra,
where they would reinforce the important notion that algebra is
about names, not letters. - Some multi-character names are permitted, as in a family such
as x1 and x2 and x3 (which do not
denote the products of x with the integers 1 and
2 and 3), and in certain function names such as
sin and cos.
Programming languages avoid such issues
very simply: every function must have a symbol, and it must be
used.
G. EMPTY VECTORS
Although MN deals with scalars, vectors,
matrices, and higher-rank arrays, it fails to provide a complete
and uniform system, particularly in the case of empty arrays.
For example, the definition of the falling factorial in Concrete
Mathematics [11] reads as x(x-1)…(x-m+1),
together with an indication that m factors are to be taken.
This pattern provides clear instructions for m down to
1, but the case of 0 requires the following note: "When m
= 0, we have = 1, because a product of no factors
is conventionally [italics added] taken to be 1 (just as
a sum of no terms is conventionally 0)."
A vector may be partitioned
by taking and dropping k items from it, and (because addition
is associative) the sum of the sums of the parts will equal the
sum over the original vector. For example:
x=: 2 3 5 7 11
k=: 3
a=: k {. x k take from x
a
2 3 5
b=: k }. x k drop from x
b
7 11
f=: +
f/a
10
f/b
18
(f/a) f (f/b)
28
f/x
28
The link function (;) boxes
its arguments and then catenates them; it can be used to display
related results compactly for comparison. For example:
k=: 3
a=: k {. x
b=: k }. x
b ; |. b
a ; b ; (f/a) ; (f/b) ; ((f/a)f(f/b)) ; (f/x)
The
case k=: 0 selects an empty vector, and the sum over it
("obviously" zero) satisfies the required identity:
k=: 0
a=: k {. x
b=: k }. x
a ; b ; (f/a) ; (f/b) ; ((f/a)f(f/b)) ; (f/x)
Because
multiplication is also associative, it satisfies a similar identity:
a ; b ; (*/a) ; (*/b) ; ((*/a)*(*/b)) ; (*/x)
Again
the function over the empty vector satisfies the identity, but
its value may no longer seem so obvious. The case of the minimum
function (also associative) may be even more surprising:
f=: <.
a ; b ; (f/a) ; (f/b) ; ((f/a)f(f/b)) ; (f/x)
Since 0+x equals x for any value of x, zero is
said to be the identity element or neutral of the
function +. Similarly, 1 is the identity element
of *, and it is clear that the result of f/ on an
empty vector must be the identity element of f if the identity
over partitions is to hold.
Why the result _ (infinity) for the case of minimum? Because
that is its identity element.
These
results for empty vectors are of practical significance, and arise
frequently. Consider, for example, the following calculation of
the falling factorial:
x=: 5
m=: 4
i. m The first m non-negative integers
0 1 2 3
x - i. m The factors of the falling factorial
5 4 3 2
*/ x - i. m The falling factorial
120
*/ x - i. 0 This correct result is the product over an empty vector
1
H. MATRIX ALGEBRA
Matrices, matrix product, and matrix inverse
are commonly introduced at high-school level and used in the treatment
of linear equations. However, we fail to exploit them in many
elementary and interesting ways. This failure rests partly on
the propensity to present significant matrices in truncated form
by suppressing (presumably for readability) zero elements.
For
example, the table of binomial coefficients introduced in Section
E appears as:
pt=: 0 1 2 3 4 !/ 0 1 2 3 4 pt 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1
but is commonly presented
as a triangle (Pascal’s triangle) by suppressing the zeros
in the lower left. These zeros are meaningful results.
For example, the number of ways of choosing 4 elements from a
list of three items (4!3) is 0.
Although Pascal’s
triangle cannot be treated as a matrix, the table pt can.
For example, its inverse (produced by the function %.)
is the table of alternating binomial coefficients:
abc=: %. pt abc 1 _1 1 _1 1 0 1 _2 3 _4 0 0 1 _3 6 0 0 0 1 _4 0 0 0 0 1
Moreover, the sums of the columns of these matrices
produce interesting results:
+/ pt 1 2 4 8 16 +/ abc 1 0 0 0 0
The "expansion" of the coefficients of a polynomial
may be expressed simply and clearly as a matrix product of the
matrix of binomial coefficients and the coefficients. For example:
X=: +/ . * Matrix product
x=: 0 1 2 3 4 5 6 Argument of polynomial
c=: 3 1 4 2 1
c p. 2 Polynomial (ascending powers)
53
c p. x
3 11 53 177 455 983 1881
d=: pt X c Expanded coefficients
d
11 19 16 6 1
d p. x Equivalent to c p. (x+1)
11 53 177 455 983 1881 3293
c p. x+1
11 53 177 455 983 1881 3293
I. TERMINOLOGY
The following is from Chapter 6 of Exploring
Math [6]:
Special terminology used in various branches of knowledge
often imposes a serious burden on a beginner. It may sometimes
be safely dismissed as pretentious and no better than familiar
terms, but serious treatment of a topic may well require finer
distinctions than those provided by familiar language. For example,
the familiar average may sometimes be substituted
for mean as defined in math and statistics.
However, mean refers not only to average (the arithmetic
mean), but also to various ways of characterizing a collection
by a single number, including the geometric mean, harmonic
mean, and common mean.
Similarly, the grammatical terms
adopted in J (from English) may seem
pretentious to anyone familiar with corresponding terms in math,
but they make possible significant distinctions that are not easily
made in MN. We illustrate this by a few sentences
and the classification of items from them in both J and
MN:
with=: & cube=: ^ with 3 commute=: ~ into=: % commute pi=: 7 into 22 2 into cube a=: i. 6
J MN Noun 22 Constant Pronoun pi Variable Verb or Function % Function or Operator Proverb cube Adverb or Operator ~ Operator Pro-adverb commute Conjunction or Operator & Operator Pro-conjunctionx with List or Vector a Vector Table or Matrix a*/a Matrix Report or Array a +/ a */ a Array
In the foregoing, MN makes the
same distinction made by noun and pronoun
in J, but uses the terms constant
and variable. The term variable may
prove somewhat misleading, particularly when used for a pronoun
such as pi (for the ratio of the circumference to the
diameter of a circle), which is not expected to vary. The
following sentences may be used to clarify the choice of the word
variable:
sqr=: *: The square function in J
(sqr 0)=(0+0)
(sqr 2)=(2+2)
(sqr 0)=(0*0)
(sqr 2)=(2*2)
(sqr 3)=(3*3)
Each of these sentences
express a "true" relation
in the sense that each comparison yields 1. However, the first
pair are true only for the specific arguments
0 and 2, and for no other. The last three suggest (correctly)
that the indicated relation remains true for any argument,
or, as we say, the argument is allowed to vary. This generality
is commonly indicated by using a pronoun argument, or,
as stated in MN, a variable:
(sqr x)=(x*x)
In MN, the term operator
(or functional) is used for both of the cases distinguished
in J by adverb and conjunction.
Moreover, in MN the term operator is also commonly
used to refer to functions.
The terms list, table,
and report are used in J with
meanings familiar to anyone, whereas the corresponding terms vector,
matrix, and array might
be known only to specialists. The familiar use of vector
is as a carrier, as in disease vector. It might be thought
that a vector "carries" its items, but
the actual etymology of the term in math
is quite different, although not as bizarre as that of matrix.
New
terminology should be approached by using dictionaries to learn
the etymology of terms, both old and new. For
example, a verb is defined as a word that (amongst other
things) expresses an action; the corresponding word function
comes from a root meaning "to perform".
Attention
to etymology is also rewarding in every-day work.
For example, the meaning of atom appears clearly
in its derivation (a[not] + tem[cut]) as something that could
not be cut.
The American Heritage Dictionary [12] presents
etymology in a particularly revealing manner:
all words derived from a given root are listed together in an
appendix. This highlights surprising and insightful relations,
such as that between tree and true.
As a further example, the root tem that occurs in atom
also occurs in anatomy, microtome, and tome.
Incidentally, tome does not necessarily mean
a big book, but rather one of the volumes "cut" from
a book, such as the 24 tomes of the original Oxford English
Dictionary.
Lewis Thomas, a noted bio-chemist,
explored the pleasure and profit of etymology
in his delightful book et cetera, et cetera. [13]. It is
well worth reading.
J. CONCLUSION
The discussion thus far has proceeded largely
by taking phrases in MN and showing how they might be re-expressed
in J. Moreover, the examples used have all been elementary,
being restricted to topics treated in high-school math, and to
an elementary subset of the facilities available in J.
The
reader may gain a deeper appreciation of the weaknesses of MN
by the converse: take statements from some of the J publications
listed in the references, and attempt to re-express them in MN,
avoiding as much as possible the use of English phrases to paper
over any deficiencies. In particular, it will be helpful to compare
the treatment in Concrete Math Companion [14] with the
conventional treatment of the same topics in Concrete Mathematics
[11].
We will illustrate the process by a few examples, showing
the complete results of all J expressions used so that
the reader may see clearly what is to be produced, even though
all details of the J expressions may not be clear. For
example:
x=: 2 3 5 7 11
+/x Sum
28
-/x Alternating sum
8
These may be re-expressed as and
, respectively. In general, the use
of -/ and the correspondingly modified matrix product -/ . *
obviates the use of the phrase .
The prefix scan adverb leads
to a more interesting case:
<x Box each prefix of x
st=: +/x Subtotals (sum each prefix)
st
2 5 10 17 28
In
MN it is usual to express the element
by using with upper and lower
limits, but it should be noted that this specifies the complete
result st only implicitly by specifying a typical element.
As shown in Section H, multiplication of a vector
of polynomial coefficients by the binomial coefficients table
pt (to be called bc here) served to produce the
expanded coefficients. The conjunction & (with)
may therefore be used to produce an expansion function exp
as follows:
X=: +/ . *
bc=: 0 1 2 3 4 !/ 0 1 2 3 4
bc
1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1
exp=: bc&X
c=: 3 1 4 2 1
x=: 0 1 2 3 4 5 6
c p. x
3 11 53 177 455 983 1881
exp c
11 19 16 6 1
(exp c) p. x
11 53 177 455 983 1881 3293
c p. x+1
11 53 177 455 983 1881 3293
exp exp c
53 73 40 10 1
(exp exp c) p. x
53 177 455 983 1881 3293 5387
c p. x+2
53 177 455 983 1881 3293 5387
exp ^: 3 c Apply exp three times
177 187 76 14 1
(exp ^: 3 c) p. x
177 455 983 1881 3293 5387 8355
c p. x+3
177 455 983 1881 3293 5387 8355
The
expansion function may also be applied to the table bc
itself, to produce successive matrix products of the binomial
coefficients table:
exp bc 1 2 4 8 16 0 1 4 12 32 0 0 1 6 24 0 0 0 1 8 0 0 0 0 1
<"2 exp^:0 1 2 3 bc
These tables show a pattern, and if we could find some
alternative expression
for each of them we would, in effect, express a host of identities
concerning the binomial coefficients. If we divide each table
by bc itself (using ordinary element-by-element division)
we do obtain a simple pattern of powers of the "order"
of each table:
<"2 (exp^:0 1 2 3 bc) %"2 bc
May
we expect programming languages to have a salutary effect on MN?
Not if it continues as de Morgan says "at the dictates
of convenience". In particular, any ideas from programming
languages would then be adopted piecemeal, further muddying the
waters.
Perhaps, on the other hand, the practical advantages of
executable notation, and the pedagogical advantages of a strict
and simple grammar, will lead to the adoption of more fundamental
and systematic change.
REFERENCES
- Cajori, F., A History of Mathematical Notations,
The Open Court Publishing Company, Chicago, 1929 (Available from
Dover). - Iverson, Kenneth E., J Introduction and Dictionary,
Iverson Software Inc. (ISI), Toronto, Ontario, 1995 - Burke, Chris,
J User Manual, ISI, 1995. - Burke, Chris, et al., J
Phrases, ISI, 1996. - Iverson, Eric, J primer, ISI,
1996 - Iverson, Kenneth E., Exploring Math, ISI, 1996.
-
Clifford A. Reiter , Fractals, Visualization and J, ISI,
1995. - Boole, G., An Investigation of the Laws of Thought,
Dover Publications, N.Y., 1951. (Originally published in 1854
by Walton and Maberly, London). - Heaviside, Oliver, Electromagnetic
Theory, Dover Publications, N.Y., 1958. (Vols 1-3 originaly
published in 1893, 1899, 1912). - Haskell B. Curry and Robert
Feys, Combinatory Logic, Volume 1, North Holland, 1974 -
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Addison Wesley, 1989. - American Heritage Dictionary of the
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