The Automath mathematics checking project and its influence on teaching: contribution to the ICMI discussion on "The influence of computers and informatics on mathematics and its teaching", Strassbourg, 25-30 March, 1985

The Automath mathematics checking project and its influence

on teaching

Citation for published version (APA):

Bruijn, de, N. G. (1985). The Automath mathematics checking project and its influence on teaching: contribution to the ICMI discussion on "The influence of computers and informatics on mathematics and its teaching", Strassbourg, 25-30 March, 1985. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8501). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1985

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providing details and we will investigate your claim.

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics

Memorandum 19.85-0 1 Issued January

1985.

Contribution to the

ICMI discussion on "The influence of computers and

informatics on mathematics and its teaching". 25-39 M a r c h 1 9 8 5 , S t r a s s b o u r g .

N.G. d e Bruijn. T H E A U T O M A T H MATHEMATICS C H E C K I N G P R O J E C T A N D I T S I N F L U E N C E O N T E A C H I N G University of Technology, Department of Mathematics, P.O. Box 513, 5600 MB Eindhoven, T h e Netherlands.

The Automath m a t h e m a t i c s c h e c k i n g p r o j e c t and i t s i n f l u e n c e on t e a c h i n g

by N. G. d e B r u i j n

1. Automath

C o m p u t e r s i n f l u e n c e m a t h e m a t i c s i n many ways. One of t h e s e l i e s i n t h e f a c t t h a t we c a n l e a r n t o e x p l a i n m a t h e m a t i c s t o a c o m p u t e r , a n d i n t h i s p r o c e s s we may l e a r n a b o u t how t o o r g a n i z e m a t h e m a t i c s and how t o t e a c h some of i t s a s p e c t s .

A t t h e T e c h n o l o g i c a l U n i v e r s i t y Eindhoven ( E i n d h o v e n , t h e N e t h e r l a n d s ) t h e p r o j e c t Automath was d e v e l o p e d from 1967 onwards, w i t h v a r i o u s k i n d s o f a c t i v i t i e s a t t h e i n t e r f a c e s of l o g i c ,

m a t h e m a t i c s , c o m p u t e r s c i e n c e , l a n g u a g e and m a t h e m a t i c a l e d u c a t i o n . R i g h t f r o m t h e s t a r t , i t was d i r e c t e d t o w a r d s t h e p r e s e n t a t i o n of knowledge b y means o f s y m b o l i c m a n i p u l a t i o n , w i t h t h e p o s s i b i l i t y t o l e a v e much o f t h e work t o a computer, w i t h q u i t e a s t r o n g e m p h a s i s on d o i n g t h i n g s i n a humanly way. One m i g h t s a y t h a t i t i s a modern v e r s i o n o f " L e i b n i z ' s dream'' of making a l a n g u a g e f o r a l l s c i e n t i f i c d i s c u s s i o n i n s u c h a way t h a t a l l r e a s o n i n g c a n be r e p r e s e n t e d by a k i n d of a l g e b r a i c m a n i p u l a t i o n .

The b a s i c i d e a o f Automath i s t h a t t h e human b e i n g p r e s e n t s a n y k i n d o f d i s c o u r s e , how l o n g i t may b e , t o a m a c h i n e , and t h a t t h e m a c h i n e c o n v i n c e s i t s e l f t h a t e v e r y t h i n g i s sound. A l l t h i s i s i n t e n d e d t o be e f f e c t i v e l y c a r r i e d o u t on a l a r g e s c a l e , and n o t j u s t " i n p r i n c i p l e " .

in any detail, but rather to explain a number of goals,

achievements and characteristics that may have a bearing on the subject of the

ICMI

discussion. The paper is definitely not trying to sell Automath as a subject to be taught to all students in standard mathematics curricula. The claim is more modest: as Automath connects so many aspects of logic, mathematics and informatics, it may be worth while to investigate whether the teaching of mathematics could somehow profit from ideas that emerged more or less naturally in the Automath enterprise. The idea of Automath is to "explain things to a machine". Students are n o machines and should be approached in a different way. But as teachers we should know that if we cannot explain a thing to a machine then we might have difficulties in explaining it to

students.

1.1.

A

basic idea of Automath is to write in the form of a complete book, line by line. A computer can check it line by line, and once that has been done, the book can be considered as mathematically correct.

1.2. As

a

starting point we think of a book written entirely by human beings. Later on we may think of leaving part of the writing to a machine. That part might be simply tedious routine work, but also possibly the more serious problem solving (i.e., "theorem proving", a branch of artificial intelligence).

1.3.

We should make a clear distinction between the Automath system and Automath books. The system consists, roughly speaking, of language rules and a computer program that checks whether any

The s y s t e m o f Automath i s m a i n l y i n v o l v e d w i t h t h e e x e c u t i o n o f s u b s t i t u t i o n , w i t h e v a l u a t i o n of t y p e s of e x p r e s s i o n s , and c o m p a r i n g s u c h t y p e s t o o n e a n o t h e r . It i s v e r y e s s e n t i a l t h a t e v e r y t h i n g t h a t i s s a i d i n a book, i s s a i d i n a p a r t i c u l a r c o n t e x t : t h e c o n t e x t c o n s i s t s o f t h e t y p e d v a r i a b l e s t h a t c a n be h a n d l e d , b u t a l s o o f t h e l i s t o f a s s u m p t i o n s t h a t c a n be u s e d . The s y s t e m k e e p s t r a c k of t h o s e c o n t e x t s .

The Automath s y s t e m d o e s n o t c o n t a i n any a p r i o r i i d e a s on what i s u s u a l l y c a l l e d l o g i c and f o u n d a t i o n of m a t h e m a t i c s . Any l o g i c a l s y s t e m ( e . g . , a n i n t u i t i o n i s t i c o n e ) c a n be

i n t r o d u c e d b y t h e u s e r i n h i s own book, and t h e same t h i n g h o l d s f o r t h e f o u n d a t i o n of m a t h e m a t i c s . I n p a r t i c u l a r , t h e u s e r i s n o t t i e d t o t h e s t a n d a r d 20-th c e n t u r y s e t t h e o r y ( Z e r m e l o - F r a e n k e l ) . And t h e u s e r c a n c h o o s e w h e t h e r t o a d m i t o r n o t t o a d m i t t h i n g s l i k e t h e axiom o f c h o i c e . From t h e n o n , t h e machine t h a t v e r i f i e s t h e u s e r ' s book w i l l be a b l e t o d o t h i s a c c o r d i n g t o t h e u s e r ' s own s t a n d a r d s .

1.4. I n a n Automath book, l o g i c and m a t h e m a t i c s a r e t r e a t e d i n e x a c t l y t h e same way. New l o g i c a l i n f e r e n c e r u l e s c a n be d e r i v e d f r o m o l d o n e s , j u s t l i k e m a t h e m a t i c a l t h e o r e m s a r e d e r i v e d , and t h e new i n f e r e n c e r u l e s c a n be a p p l i e d a s l o g i c a l t o o l s , i n t h e same way a s m a t h e m a t i c a l t h e o r e m s a r e a p p l i e d . 1.5. W r i t i n g i n Automath c a n be t e d i o u s . A l l d e t a i l s of a r g u m e n t s h a v e t o be p r e s e n t e d most m e t i c u o u s l y . A t f i r s t s i g h t t h i s m i g h t be v e r y i r r i t a t i n g . The q u e s t i o n s a r e ( i ) whose f a u l t t h i s i s , and ( i i ) what c a n be done a b o u t i t .

that the length of an Automath book makes, is due to the fact that no attempt was made to "do something about it" at the stage of the design

of

the general system. This is based on the

philosophy that generality comes first, and that adaptability to special situations is a second concern.

The reason why Automath books become so long is that we claim to be able to handle all usual mathematical discourse, but the mathematician has more in his mind than he explains. Perhaps we may say that part of mathematical work is done subconciously. Mathematicians have a vast "experience" in mathematical situations, and such experience may give a strong feeling for how all the little gaps can be filled. Possibly much of the experience is consulted subconciously "on the spot".

Moreover, mathematical talking and writing are social

activities. In every area, people talk and write in a style they know they can get away with. Some poor or incomplete forms of discourse are so wide-spread that it seems silly to bother about improvements; certainly it is not a very rewarding task to try.

The answer to question (ii) is that very much can be done about it indeed. But just like every user can write his own book under the Automath system, he can implement his own attachments to the system. This may involve special

abbreviation facilities, but also automatized text writing, producing packages of Autonath lines by means of a single command, in cases where there is a clear system behind such a package.

1.6.

Are computers essential for Autonath? Not absolutely. The computer sets the standard for what the notion

v e r i f y m a t h e m a t i c a l d i s c o u r s e , we h a v e n o t p r o p e r l y f o r m a l i z e d i t y e t . I n t h e s t a n d a r d f o r m , t h e a u t h o r o f an Automath book h a s t o w r i t e a l l t h e symbols one by o n e , and s i n c e he knows t h a t what he w r i t e s i s c o r r e c t , he would a l s o be a b l e t o c h e c k i t by hand.

N e v e r t h e l e s s humans make m i s t a k e s . Automath books have

been w r i t t e n w i t h a number of c h a r a c t e r s of t h e o r d e r of a n i l l i o n , a l l t y p e d by hand. It i s h a r d t o g u a r a n t e e c o r r e c t n e s s of s u c h a t e x t w i t h o u t t h e h e l p o f a modern c o m p u t e r . 1.7. A s t h e Automath s y s t e m h a s no a p r i o r i knowledge of l o g i c and s e t t h e o r y , i t c a n be u s e d t o w r i t e i n a s t y l e t h a t m i g h t be more n a t u r a l t h a n what we s e e i n o t h e r f o r m a l i z a t i o n s . T h e r e i s a wide-spread i d e a t h a t p r o p o s i t i o n a l l o g i c comes down t o m a n i p u l a t i n g f o r m u l a s i n a b o o l e a n a l g e b r a , a k i n d of m a n i p u l a t i o n t h a t i s e i t h e r c a r r i e d o u t by h a n d l i n g f o r m u l a s w i t h t h e a i d of l i s t s o f t a u t o l o g i e s ( i n t h e same way a s one used

t o d o i n t r i g o n o m e t r y ) , o r by a machine t h a t c h e c k s a l l p o s s i b i l i t i e s of z e r o s and o n e s a s v a l u e s f o r t h e b o o l e a n

v a r i a b l e s . A v e r y much b e t t e r f o r m a l i z a t i o n l i e s i n t h e s y s t e m o f " n a t u r a l d e d u c t i o n " . T h i s i s v e r y e a s y i n Automath. The

b o o l e a n b i t - h a n d l i n g p r o p o s i t i o n a l l o g i c c a n be done i n Automath t o o , b u t i t i s much more clumsy t h a n n a t u r a l d e d u c t i o n .

A s e c o n d o p t i o n we g e t from t h e l i b e r t y of u s i n g Automath i n t h e s t y l e we p r e f e r , i s t o g i v e up t h e 20-th c e n t u r y i d e a t h a t " e v e r y t h i n g i s a s e t " . T h e r e i s t h e magic Zermelo-Fraenkel u n i v e r s e i n w h i c h e v e r y p o i n t i s a s e t , and somehow a l l m a t h e m a t i c a l o b j e c t s a r e t o be coded a s p o i n t s i n t h a t u n i v e r s e . The p a r t i c u l a r c o d i n g i s a m a t t e r of f r e e c h o i c e : t h e r e i s no n a t u r a l way t o c o d e . Z e r m e l o - F r a e n k e l s e t t h e o r y i s q u i t e a heavy m a c h i n e r y t o be

t a k e n a s a b a s i s f o r m a t h e m a t i c s , and n o t many m a t h e m a t i c i a n s a c t u a l l y know i t . An a l t e r n a t i v e i s t o t a k e " t y p e d s e t t h e o r y " , i n w h i c h t h i n g s a r e c o l l e c t e d t o s e t s o n l y i f t h e y a r e o f t h e same t y p e : s e t s o f n u m b e r s , s e t s of l e t t e r s , s e t s of t r i a n g l e s , e t c . It may t a k e some t r o u b l e t o make u p one's mind a b o u t t h e q u e s t i o n what b a s i c r u l e s f o r t y p e d s e t t h e o r y s h o u l d be t a k e n a s p r i m i t i v e s , b u t i f we j u s t s t a r t t a l k i n g t h e way we d i d m a t h e m a t i c s b e f o r e modern s e t t h e o r y e m e r g e d , we s e e t h a t we need v e r y l i t t l e . Anyway, i n Automath we h a v e n o t r o u b l e a t a l l t o t a l k m a t h e m a t i c s i n a sound o l d - f a s h i o n e d way. Y e t , i f someone s t i l l w a n t s t o t a l k i n t e r m s of Zermelo- F r a e n k e l u n i v e r s e , Automath i s r e a d y t o t a k e i t . 1.8. One o f t h e a d v a n t a g e s of Automath n o t b e i n g t i e d t o a n y p a r t i c u l a r s y s t e m f o r l o g i c and s e t t h e o r y , i s t h a t we c a n t h i n k o f f o r m a l i z i n g e n t i r e l y d i f f e r e n t t h i n g s t o o , a g a i n i n a n a t u r a l s t y l e . A s a n example we may t h i n k of t h e a l g o r i t h m i c d e s c r i p t i o n of g e o m e t r i c a l c o n s t r u c t i o n s l i k e t h o s e w i t h r u l e r and compass. A l t h o u g h i t h a s n o t a c t u a l l y been p r o d u c e d , we may t h i n k of a s i n g l e Automath book

c o n t a i n i n g l o g i c , m a t h e m a t i c s and t h e d e s c r i p t i o n of r u l e r and compass c o n s t r u c t i o n s , w i t h i n p a r t i c u l a r t h e d e s c r i p t i o n and c o r r e c t n e s s p r o o f ( b o t h due t o Gausz) of t h e c o n s t r u c t i o n of t h e r e g u l a r 17-gon. T h i s d e s c r i p t i o n w i l l be q u i t e d i f f e r e n t from c o d i n g t h e c o n s t r u c t i o n a s a p o i n t i n t h e Zermelo-Fraenkel u n i v e r s e . We m i g h t e v e n t h i n k of a r o b o t e q u i p p e d w i t h r u l e r , c o m p a s s , p e n c i l and p a p e r , who r e a d s t h e d e t a i l s of t h e c o n s t r u c t i o n f r o m t h e Automath book and c a r r i e s them o u t i n t h e way Gausz m e a n t .

of theory, connected by rather vague intuitive ideas. Ever since the last part of the 19-th century it has been one of the ideas of the mathematical community that mathematics should be

integrated: all parts of mathematics are to become sub-domains of one single big theory. The patchwork picture still applies to most physical sciences, but also to several parts of the mathematical sciences. One such part is informatics.

It seems to be a good idea to integrate informatics into mathematics, at least in principle. And, as in the case of geometrical constructions, Automath is a good candidate for describing this. It is possible to write an Automath book containing: logic, mathematics, description of syntax and semantics of a programming language, and particular programs with proofs that the execution achieves the solution of particular mathematical problems. One might even think of going further: description of the computer hardware with proof that it garantees the realization of the programming language semantics. Or directly, without the intervention of a

programming language, that a given piece of hardware produces a result with a given mathematical specification.

Needless to day, this kind of integrated theory will always contain a number of primitives we have no proof for, but it will be absolutely clear in the Automath book what these primitives are.

1.10.

One thing people like in Automath, and other people strongly dislike, is the way Automath treats proofs as if they were mathematical objects. This is called "propositions as types". As the type of a proof we have something that is

One should not be worried about this. Automath does not say that proofs are objects, but just treats them syntactically in the same way as objects are treated. This turns out to be very profitable: it simplifies the system, as well as its

language theory and the computer verification of books. A third case where things are treated as objects is the one of the geometrical constructions we mentioned in

1.8.

1.11. In standard mathematics, most identifiers are letters of various kinds, possibly provided with indices, asterisks and the like. And then there are the numerals, of course. We have learned from programming languages, however, to use arbitrary combinations of letters and numerals as identifiers, (with restrictions

like not to begin with a numeral). We do the same thing in Automath, thus having the possibility to choose identifiers with a mnemonic value, like "~essel", "Theoreml37",

I I

commutative". This certainly helps to keep books readable. In contrast to programming languages, the Automath system does not have the numerals 0,1,...,9. One can introduce them

as identifiers in a book containing the elements of natural number theory, taking "0" and "succ" (for "successor") as primitive, and defining l:=succ(O), 2:=succ(l),..., 9:=succ(8), ten:=succ(9). After having introduced addition and multiplication, we can define things like thirtyseven:=sun(prod(3,ten),7),

but the Automath system has no facilities to write this as 37. This decimal notation might be added as an extra (it is one of the possible "attachments" mentioned in 1.5).

1.12. One of the basic aims of the Autonath enterprise was to keep it feasible. This has been achieved indeed: considerable

p o r t i o n s o f m a t h e m a t i c s o f v a r i o u s k i n d s h a v e been " t r a n s l a t e d " i n t o Automath, and t h e e f f o r t needed f o r t h i s r e m a i n e d w i t h i n r e a s o n a b l e l i m i t s . I f we s t a r t from a p i e c e of m a t h e m a t i c s t h a t i s sound and w e l l u n d e r s t o o d , i t can be t r a n s l a t e d . It may a l w a y s t a k e some t i m e t o d e c i d e how t o s t a r t , b u t i n t h e l o n g r u n t h e t r a n s l a t i o n i s a m a t t e r of r o u t i n e . A s a r u l e of thumb we may s a y t h e r e i s a l o s s f a c t o r of t h e o r d e r o f 10: i t t a k e s a b o u t t e n t i m e s a s much s p a c e and t e n t i m e s a s much t i m e a s w r i t i n g m a t h e m a t i c s t h e o r d i n a r y way. But i t i s n o t o v e r i m p o r t a n t how b i g t h i s l o s s f a c t o r i s ( i t would n o t be h a r d t o r e d u c e i t by means of s u i t a b l e a t t a c h m e n t s , a d a p t e d t o t h e n a t u r e of t h e s u b j e c t m a t t e r ) . What r e a l l y m a t t e r s i s t h a t i t d o e s n o t t e n d t o i n f i n i t y , w h i c h h a p p e n s i n many o t h e r s y s t e m s o f f o r m a l i z i n g m a t h e m a t i c s . The main r e a s o n f o r t h e l o s s f a c t o r b e i n g c o n s t a n t i s t h a t Automath h a s t h e same f a c i l i t i e s f o r u s i n g d e f i n i t i o n s ( w h i c h a r e , e s s e n t i a l l y , a b b r e v i a t i o n s ) a s one h a s i n s t a n d a r d m a t h e m a t i c s . The f a c t t h a t t h e s y s t e m of r e f e r e n c e s i s s u p e r i o r t o what we h a v e i n s t a n d a r d m a t h e m a t i c s , makes i t p o s s i b l e t h a t t h e l o s s f a c t o r e v e n d e c r e a s e s on t h e l o n g r u n when d e a l i n g w i t h a l a r g e book. 1.13. A n o t h e r f e a t u r e t h a t makes Automath f e a s i b l e i s t h a t we n e e d n o t a l w a y s s t a r t a t t h e b e g i n n i n g : we can s t a r t somewhere i n t h e m i d d l e , and i f we need s o m e t h i n g t h a t we have n o t d e f i n e d , o r h a v e n o t p r o v e d , we j u s t t a k e i t a s a p r i m i t i v e ( p r i m i t i v e n o t i o n o r a x i o m ) and we g o on. We c a n l e a v e i t t o l a t e r a c t i v i t y t o r e p l a c e a l l t h e s e p r i m i t i v e s by d e f i n e d o b j e c t s and p r o v e n t h e o r e m s . T h i s k i n d of t a c t i c s was o f t e n ( a b o u t 30 c a s e s ) a p p l i e d a t Eindhoven by s t u d e n t s ( m a t h e m a t i c s m a j o r s ) . I t u s u a l l y t o o k

t h e s t u d e n t n o t much more t h a n 100 h o u r s work t o l e a r n a b o u t t h e s y s t e m , t o t r a n s l a t e a g i v e n p i e c e of m a t h e m a t i c s , t o u s e t h e c o n v e r s a t i o n a l f a c i l i t i e s a t a computer t e r m i n a l , and t o f i n i s h w i t h a c o m p l e t e l y v e r i f i e d Automath book c o n t a i n i n g t h e r e s u l t . I n o r d e r t o g i v e a n i d e a o f t h e s u b j e c t s t h a t had t o be t r a n s l a t e d we m e n t i o n a few: ( i ) The W e i e r s t r a s z t h e o r e m t h a t s a y s t h a t t h e t r i g o n o m e t r i c p o l y n o m i a l s l i e d e n s e i n t h e s p a c e o f c o n t i n u o u s p e r i o d i c f u n c t i o n s , ( i i ) The B a n a c h - S t e i n h a u s t h e o r e m , ( i i i ) The f i r s t e l e m e n t s of g r o u p t h e o r y .

1.14. Of t h e more e x t e n s i v e books t h a t were w r i t t e n i n Automath we m e n t i o n two. The f i r s t o n e i s L.S. J u t t i n g ' s c o m p l e t e

t r a n s l a t i o n o f E. Landau's Grundlagen d e r A n a l y s i s . I n o r d e r t o t e s t t h e f e a s i b i l i t y of t h e s y s t e m , t h e t r a n s l a t o r k e p t

h i m s e l f s t r i c t l y t o Landau's t e x t , r a t h e r t h a n i n v e n t i n g some o f the many p o s s i b l e s h o r t c u t s and improvements t h a t would make t h e t r a n s l a t i o n e a s i e r and s h o r t e r . The second o n e w e m e n t i o n h e r e was by J.T. Udding, who w r o t e a new t e x t w i t h a b o u t t h e same r e s u l t s , much b e t t e r s u i t e d t o t h e

Automath s y s t e m , b o t h i n i t s g e n e r a l o u t l i n e and i n

I t s d e t a i l s . The g a i n o v e r Landau's t e x t , i n s p a c e a s w e l l a s i n t i m e , was r o u g h l y 2.5.

1.15. One o f t h e i d e a s of t h e Automath e n t e r p r i s e was t o g e t e v e n t u a l l y t o a b i g m a t h e m a t i c a l e n c y c l o p a e d i a , a d a t a b a n k , c o n t a i n i n g a v a s t p o r t i o n o f m a t h e m a t i c s i n a b s o l u t e l y

d e p e n d a b l e form. T h i s i s a t h i n g t h a t would t a k e many h u n d r e d s o f man y e a r s ( t h u s f a r t h e A u t o n a t h p r o j e c t t o o k s o m e t h i n g l i k e 4 0 ) . But t h e i d e a i s f e a s i b l e . Most of t h e s t u d e n t s m e n t i o n e d i n 1.13 u s e d t h e Landau t r a n s l a t i o n ( s e e 1.14) a s a d a t a

-

11

-

b a n k , and t h a t way t h e y added t o t h e bank.

2. S t a n d a r d m a t h e m a t i c a l l a n g u a g e . I n c l o s e c o n n e c t i o n w i t h Automath a l a n g u a g e was s t u d i e d w i t h t h e same l e v e l of p r e c i s i o n , b u t c l o s e r t o o r d i n a r y l a n g u a g e a s w r i t t e n by m a t h e m a t i c i a n s , a t l e a s t when t h e y a r e v e r y p r e c i s e . L e t u s c a l l i t

MV

( f o r " m a t h e m a t i c a l v e r n a c u l a r " ) .

MV

i s t h e f a m i l i a r m i x t u r e of words and f o r m u l a s i n which some o f t h e l e t t e r s and f o r m u l a s p l a y a s y n t a c t i c r o l e j u s t a s i f t h e y were o r d i n a r y p a r t s of a s e n t e n c e , l i k e s u b j e c t , d i r e c t o b j e c t , e t c .

2.1. I t i s p o s s i b l e t o f o r m u l a t e l o g i c and t h e f o u n d a t i o n of

m a t h e m a t i c s i n t e r m s of t h e grammar of s u c h a l a n g u a g e . The grammar of MV c a n be k e p t q u i t e s i m p l e , s i n c e a l l s o r t s of i d i o m of n a t u r a l l a n g u a g e c a n be c a u g h t i n t e r m s o f d e f i n i t i o n s i n t h e book. T h i s way we do n o t need t o d i s t i n g u i s h more t h e n t h e f o l l o w i n g f o u r g r a m m a t i c a l c a t e g o r i e s : ( i ) s e n t e n c e s , ( i i ) s u b s t a n t i v e s , ( i i i ) names, ( i v ) a d j e c t i v e s . Each one of t h e s e f o u r c a n o c c u r a s a g r o u p of w o r d s , b u t a l s o a s a m a t h e m a t i c a l symbol, a f o r m u l a , o r a

m i x t u r e o f w o r d s and f o r m u l a s . The f o u r c a t e g o r i e s c o r r e s p o n d t o t h e f o u r k i n d s o f d e f i n i t i o n s t h a t m a t h e m a t i c i a n s g i v e . I n

t h e d e f i n i t i o n s o f t h e f i r s t k i n d t h e new t e r m i s a s e n t e n c e (1ike:"we s a y t h a t p d i v i d e s q i f

..."),

i n t h e second c a s e i t i s a s u b s t a n t i v e ("a s q u a r e i s a ...I1), i n t h e t h i r d one a name (.

..

i s c a l l e d t h e n-th B e s s e l c o e f f i c i e n t ) , i n

2.2. The d i f f e r e n c e i n s y n t a x i s n o t t h e o n l y d i f f e r e n c e

b e t w e e n Automath and MV. The main d i f f e r e n c e i s t h a t i n Automath

I e a c h l i n e c o n t a i n s e x a c t l y a l l i n f o r m a t i o n a b o u t how t h e s t a t e d r e s u l t f o l l o w s f r o m p r e v i o u s l i n e s : a l l t h e o r e m s and i n f e r e n c e r u l e s w h i c h a r e u s e d a r e m e n t i o n e d , and t h e i r r o l e i s made a b s o l u t e l y c l e a r . I n MV s u c h i n d i c a t i o n s do n o t b e l o n g t o t h e l a n g u a g e i t s e l f , b u t c a n be c o n s i d e r e d a s h a v i n g been w r i t t e n i n t h e m a r g i n . I n o t h e r w o r d s , i n Automath t h e y a r e l a n g u a g e , i n MV m e t a l a n g u a g e . One c a n u s e MV as a s t a g e i n t h e p r o c e s s of w r i t i n g i n Automath. I f t h e s t e p s i n PN a r e s m a l l , and i f t h e i n d i c a t i o n s i n t h e m a r g i n a r e s u f f i c i e n t l y c l e a r , t h e t r a n s l a t i o n i n t o Automath i s a r o u t i n e m a t t e r . 2.3. I n s p e c t i n g t e x t b o o k s i n m a t h e m a t i c s on s c h o o l l e v e l o n e f i n d s v e r y l i t t l e

MV.

Most of t h e t e x t s a r e w r i t t e n i n m e t a l a n g u a g e s o f v a r i o u s k i n d s . Q u i t e o f t e n , t h e i n t e r s e c t i o n o f t h e t e x t w i t h i t s own r e p r e s e n t a t i o n i n MV i s l i t t l e more t h a n t h e m a t h e m a t i c a l f o r m u l a s , i . e . t h e p a r t t h a t was f o r m a l i z e d h u n d r e d s o f y e a r s a g o . 3 . E f f e c t s on m a t h e m a t i c a l e d u c a t i o n .

The q u e s t i o n was: "How do c o m p u t e r s and i n f o r m a t i c s i n f l u e n c e m a t h e m a t i c a l i d e a s , v a l u e s and t h e advancement of m a t h e m a t i c a l s c i e n c e ? " . T h e r e w i l l be a l l s o r t s of i n f l u e n c e s , l i k e t h e t a s t e f o r c o n s t r u c t i v i t y , a n d , a s f a r a s e d u c a t i o n i s c o n c e r n e d , t h e new p o s s i b i l i t i e s t o l e t s t u d e n t s h a v e t h e i r own s t i m u l a t i n g d i s c o v e r i e s w i t h t h e a i d o f a c o m p u t e r . B u t

the influence we get from the fact that we can explain

mathematics to a computer, should not be forgotten. We shall look into this in some detail.

3.1.

First, there are the philosophic aspects. Is it really mathematics we explain to a computer? Or is it just some piece of code we happen to interpret as mathematics? How arbitrary is our interpretation?

There is no definite answer to such questions.

If

we have to compare a formal system to something that is partly intuitive, then the comparison cannot be completely formal.

For example, in the partially intuitive mathematical world, the question whether the mathematical objects exist in a platonic reality, might seem to make some philosophical sense. But if we consider a completely formalized version to be explained to a computer, such a question cannot even be formulated. Some people will react by saying that this definitely puts an end to platonism, others will say that it shows that no formalization will ever be complete

.

3.2.

Having to phrase our mathematics in a very definite language, we have to make clear what part of ordinary mathematics belongs to the language and what part is metalanguage. Many paradoxes arise just by confusing language and metalanguage. Making the distinction will certainly help to understand mathematics better.

3.3. Today, most mathematicians have the idea that the foundation of mathematics is too hard to learn for a

non-specialist, and can only be taught to students who know mathematics already. This means that the foundations of the

b u i l d i n g o f m a t h e m a t i c s a r e l a i d o n l y a f t e r t h e b u i l d i n g i s c o m p l e t e d , s o t h e y c a n i m p o s s i b l y p l a y t h e r o l e of t h e b a s i s o f m a t h e m a t i c s . The t e a c h i n g of t h e f o u n d a t i o n s a t t h a t l a t e s t a g e a s s u m e s t h e s t u d e n t s t o be a c q u a i n t e d w i t h m a t h e m a t i c a l i d e a s ( t h e r o l e of d e f i n i t i o n s , a x i o m s , t h e o r e m s ) f o r w h i c h o n e e x p e c t s t h e f o u n d a t i o n s t o g i v e e x p l a n a t i o n s . On a l o w e r l e v e l , t h e same t h i n g h a p p e n s i n t h e b o o l e a n p r o p o s i t i o n a l c a l c u l u s : i t i s a m a t h e m a t i c a l s y s t e m w h i c h i s e r e c t e d by s t a n d a r d m a t h e m a t i c a l t e c h n i q u e s , a n d n e v e r t h e l e s s i t i s a p o p u l a r b e l i e f t h a t i t c a n e x p l a i n w h a t l o g i c i s , what p r o o f s a r e . 3 . 4 . O u t s i d e r s would be v e r y s u r p r i s e d t o h e a r t h a t m a t h e m a t i c i a n s a r e s o v a g u e a b o u t t h e i r own f o u n d a t i o n s , e v e n now, t o w a r d s t h e end of t h e 20-th c e n t u r y , t h a t g r e a t c e n t u r y f o r l o g i c . I f o n e r e a l l y t a k e s t h e t a s k s e r i o u s l y t o w r i t e ( l i k e i t c a n be done i n Automath) t h e f o u n d a t i o n s of m a t h e m a t i c s up t o a l e v e l s u c h t h a t t h e w o r k i n g m a t h e m a t i c i a n would be a b l e t o b u i l d on i t , o n e w i l l s e e t h a t i t i s n o t a t a l l t h a t h a r d . A sound b a s i s c a n e a s i l y be g i v e n a t t h e a g e of 1 7 t o 19. F o r many q u e s t i o n s a b o u t t h e r e l a t i o n between m a t h e m a t i c s and c o m p u t e r s ( q u e s t i o n s l i k e program c o r r e c t n e s s ) i t i s v e r y e s s e n t i a l t o h a v e s u c h a b a s i s .

Of c o u r s e , t h e b a s i s need n o t be g i v e n i t s e l f i n a f o r m a l l a n g u a g e . I t c a n be q u i t e i n f o r m a l , b u t t h e t e a c h e r s h o u l d know t h e f o r m a l b a c k g r o u n d .

The method o f n a t u r a l d e d u c t i o n i s a v e r y good c a n d i d a t e f o r e x p l a i n i n g t h e f o u n d a t i o n o f m a t h e m a t i c s . It opens t h e p o s s i b i l i t y t o t r e a t t h e i n t r o d u c t i o n and e l i m i n a t i o n r u l e s of t h e p r o p o s i t i o n a l c a l c u l u s i n e x a c t l y t h e same s t y l e a s t h o s e of t h e p r e d i c a t e

c a l c u l u s . Moreover, i t c a n be p o i n t e d o u t t o t h e s t u d e n t , by means o f a n i n f o r m a l m e t a l a n g u a g e , what i s a p r o o f , a n axiom, a d e f i n i t i o n , a n a s s u m p t i o n , a t h e o r e m . And i t o p e n s t h e way t o u n d e r s t a n d i n g n o t i o n s t h a t c a n n o t be p r o p e r l y e x p l a i n e d a t a l l on a n i n f o r m a l b a s i s . I n t h i s c o n n e c t i o n we m e n t i o n t h e n o t i o n o f e x i s t e n c e , w h i c h h a s r e m a i n e d a m y s t e r y t o many g e n e r a t i o n s of m a t h e m a t i c i a n s . 3.5. A f o u n d a t i o n s c o u r s e a t a n e a r l y s t a g e s h o u l d be recommended. T h i s i s n o t o n l y b e c a u s e of t h e c o m p u t e r ; a n o t h e r i m p o r t a n t r e a s o n i s t h e d i s i n t e g r a t i o n of t h e t e a c h i n g o f g e o m e t r y . T r a d i t i o n a l l y , s c h o o l g e o m e t r y used t o g i v e t h e i n i t i a t i o n i n t o m a t h e m a t i c a l r e a s o n i n g . O t h e r m a t h e m a t i c a l s u b j e c t s u s e d t o t r a i n t h e a r t of c a l c u l a t i o n , n o t t h e a r t of p r o o f . But g e o m e t r y had i t s d r a w b a c k s : i t was h a r d and u n a t t r a c t i v e t o k e e p t h e r e a s o n i n g p u r e , i . e . , t o remove e v e r y a p p e a l t o what we l e a r n by o b s e r v a t i o n o f t h e p h y s i c a l w o r l d . I n p a r t i c u l a r t h i s r e f e r s t o t h e m a t t e r of o r d e r on t h e l i n e and i n t h e p l a n e . A n o t h e r drawback was t h a t q u i t e o f t e n t h e a r g u m e n t s f a i l e d i n some e x c e p t i o n a l , o f t e n t r i v i a l , s i t u a t i o n s , and t h a t t h e s e had t o be t r e a t e d s e p a r a t e l y . And a s a t i s f a c t o r y t r e a t m e n t o f t h e a x i o m a t i c b a s i s was t o o d i f f i c u l t t o be t r e a t e d a t s c h o o l . And, l a s t l y , t h e l o g i c a l c o n t e n t was s o l i m i t e d : n o p r e d i c a t e c a l c u l u s , n o q u a n t i f i e r s , a p a r t from a few c a s e s where s e t s p l a y e d a r o l e ( t h e g e o m e t r i c l o c i ) . On t h e o t h e r h a n d , g e o m e t r y showed a w o n d e r f u l i n t e r p l a y between i n t u i t i o n and a r g u m e n t a t i o n .

P o s s i b l y b e c a u s e of t h e drawbacks m e n t i o n e d h e r e , t r a d i t i o n a l s c h o o l g e o m e t r y was a l m o s t e n t i r e l y d i s c a r d e d i n most c o u n t r i e s , and r e p l a c e d by t h e s t u d y of " s t r u c t u r e s " , c a l l e d "new math". I n t h e s e new s u b j e c t s t h e r e was h a r d l y a c h a n c e t o t r a i n t h e a r t of

p r o o f , and now we a r e l e f t w i t h t h e s a d s i t u a t i o n t h a t upon e n t r y o f t h e u n i v e r s i t y t h e s t u d e n t s , e v e n m a t h e m a t i c s and c o m p u t e r s c i e n c e m a j o r s , a r e v e r y weak i n t h i s r e s p e c t . 3 . 6 . I n many p a r t s of t h e new m a t h , i n p a r t i c u l a r i n a l g e b r a i c a r e a s , i t i s q u i t e h a r d t o draw t h e b o a r d e r l i n e b e t w e e n m a t h e m a t i c s a n d m e t a m a t h e m a t i c s ( c f . 2 . 3 ) . And r e a s o n i n g a b o u t s e t s , w i t h o r w i t h o u t Venn d i a g r a m , i s o f t e n on a low l o g i c a l l e v e l . I n p a r t i c u l a r , i t g i v e s h a r d l y a n y o p p o r t u n i t y f o r h a n d l i n g v a r i a b l e s . I t h a s t o be t a d m i t t e d t h a t t h e i n n o v a t i o n s i n m a t h e m a t i c a l e d u c a t i o n have g i v e n u s q u i t e some p r o g r e s s , b o t h i n i n s i g h t s a s i n p r a c t i c a l a p p l i c a b i l i t y , b u t t h e p r i c e we p a i d by n e g l e c t i n g t h e a r t of p r o o f may h a v e been t o o h i g h . 3.7. M a t h e m a t i c s m a j o r s on t h e u n i v e r s i t y l e v e l u s u a l l y l e a r n t o h a n d l e p r e d i c a t e c a l c u l u s i n c o u r s e s on t h e f o u n d a t i o n o f a n a l y s i s . A t l e a s t t h e y l e a r n i t i m p l i c i t l y , on a p r a c t i c a l b a s i s , and d i r e c t l y t i e d t o t h e f o r m a l i z a t i o n of n o t i o n s w i t h a n i n t u i t i v e background, l i k e u n i f o r m c o n v e r g e n c e . N e e d l e s s t o s a y t h i s k i n d of m a t e r i a l w i l l become g r a d u a l l y h a r d e r now t h a t t h e s t u d e n t s e n t e r t h e u n i v e r s i t y w i t h s u c h a p o o r p r e p a r a t i o n i n t h e a r t of p r o o f . A n o t h e r m a t t e r i s t h a t i t i s n o l o n g e r c l e a r w h e t h e r i n f o r m a t i c s s t u d e n t s s h o u l d t a k e c o u r s e s i n t h e f o u n d a t i o n o f a n a l y s i s . T h e r e i s a d a n g e r t h a t i n t h e n e a r f u t u r e t h e o n l y i n t e r s e c t i o n o f t h e c u r r i c u l a f o r m a t h e m a t i c s and i n f o r m a t i c s w i l l be some k i n d of s i m p l e c a l c u l u s . 3 . 8 . A s t o t e a c h i n g t h e a r t o f p r o o f , i t may be a good

i d e a n o t t o t i e i t t o g e o m e t r y , and n o t t o any new s u b j e c t l i k e c o m b i n a t o r i c s , s e t t h e o r y o r a l g e b r a , b u t t o t a k e i t a s a s u b j e c t i n i t s own r i g h t , i n t h e form of a n e l e m e n t a r y l o g i c s c o u r s e . A s a k i n d o f e x p e r i m e n t s u c h a c o u r s e was t r i e d f o r computer s c i e n c e s t u d e n t s , r i g h t f r o m s c h o o l , a t t h e T e c h n o l o g i c a l

U n i v e r s i t y E i n d h o v e n s i n c e 1982. It seems t o have been s u c c e s s f u l i n t e a c h i n g t h e s t r u c t u r e of p r o o f by means of e x p l a i n i n g t h e r u l e s o f t h e game o f p r o p o s i t i o n a l and p r e d i c a t e c a l c u l u s , The b a s i s was n a t u r a l d e d u c t i o n ( c f . 3 . 4 ) . Only a f t e r t h e b u i l d i n g o f l o g i c was e r e c t e d , i t was shown how t h e n o t i o n of v a l u a t i o n g i v e s t h e l i n k w i t h t h e b o o l e a n a l g e b r a a s p e c t , The c o u r s e s t a r t e d w i t h a c h a p t e r on s y n t a x , i n v o l v i n g t h e s t u d y of p a r e n t h e s e s , r e p r e s e n t a t i o n of f o r m u l a s as t r e e s , i n f i x n o t a t i o n , bound v a r i a b l e s , lambda c a l c u l u s n o t a t i o n , s u b s t i t u t i o n , e t c . I t t u r n e d o u t t o be i l l u m i n a t i n g t o t a k e t h e t r e e s a s t h e c e n t r a l t h e m e , i n p a r t i c u l a r i n c o n n e c t i o n w i t h s u b s t i t u t i o n s i n f o r m u l a s w i t h bound v a r i a b l e s . I n t h e t r e a t m e n t o f p r e d i c a t e l o g i c , p r e d i c a t e s were t a k e n t o be d e f i n e d on s e t s , and i n t h a t r e s p e c t t h e c o u r s e t o o k a n a i v e p o i n t o f view. It was n o t a t t e m p t e d t o d e v e l o p t h e l a n g u a g e o f m a t h e m a t i c s i n a l l i t s g l o r y : t h a t would p r o b a b l y have t a k e n t w i c e a s much t i m e a s c o u l d r e a s o n a b l y be d e v o t e d t o t h e c o u r s e . T h i s i n t r o d u c t o r y c o u r s e on l o g i c t o o k n o t more t h a n 18 h o u r s t e a c h i n g , w i t h a b o u t 24 h o u r s added f o r e x c e r c i s e s . I n a s e q u e l of t h i s c o u r s e ( a g a i n 18 h o u r s t e a c h i n g p l u s e x c e r c i s e s ) , a p p l i c a t i o n s were made t o m a t h e m a t i c a l f u n d a m e n t a l s ( t r e a t m e n t of s e t s and m a p p i n g s , t h e s y s t e m of n a t u r a l numbers, t h e method o f i n d u c t i o n , r e c u r s i o n and d e f i n i t i o n by r e c u r s i o n ) , b u t a l s o t o a number of s u b j e c t s on t h e b o a r d e r l i n e of n a t h e m a t i c s

and i n f o r m a t i c s . T h e s e were m a i n l y : t h e t e r m i n o l o g y of t h e f r e e monoid a n d i t s r e l a t i o n t o l a n g u a g e , c o n t e x t f r e e grammars i n a m a t h e m a t i c a l s e t t i n g ( w i t h t e r m i n a l s and n o n - t e r m i n a l s ) , a n d t h e r e l a t i o n of t h i s w i t h t h e Backus-Naur form. 3 . 9 . A c o u r s e l i k e t h e one d e s c r i b e d i n 3.8 m i g h t be recommended a s t h e body of t h e i n t e r s e c t i o n of t h e c u r r i c u l a of m a t h e m a t i c s and i n f o r m a t i c s . What m i g h t be added t o t h e i n t e r s e c t i o n i s a m a t h e m a t i c a l d e s c r i p t i o n o f what i s a c o m p u t e r , a p r o g r a m , i n p u t , o u t p u t , p r o g r a m s p e c i f i c a t i o n and program c o r r e c t n e s s . A t t h a t s t a g e i t i s b e t t e r n o t t o go i n t o d e t a i l s of a programming l a n g u a g e , a p a r t f r o m t h e d e s c r i p t i o n how s u c h l a n g u a g e s c a n be d e f i n e d by r e c u r s i o n . 3.10. P a r t s o f t h e l o g i c s c o u r s e , l i k e s y n t a x and p r o p o s i t i o n a l c a l c u l u s i n n a t u r a l d e d u c t i o n , might be s h i f t e d t o t h e s c h o o l a g e (16-18 y e a r s ) . The n a t u r a l d e d u c t i o n would be v e r y a p p r o p r i a t e f o r showing what a p r o o f i s , a n d i t would r a i s e t h e t e a c h i n g of l o g i c above t h e " t r i g o n o m e t r y l e v e l f ' ( c f . 1.7). And lambda c a l c u l u s m i g h t r e a l l y h e l p t o make s c h o o l m a t h e m a t i c s e a s i e r . 3.11. Some o f t h e m a t e r i a l m e n t i o n e d i n s e c t i o n 2 was t a u g h t a t E i n d h o v e n s i n c e a b o u t 1977 i n a c o u r s e c a l l e d "Language a n d s t r u c t u r e of m a t h e m a t i c s " , f o r t h o s e m a t h e m a t i c s m a j o r s who wanted a t e a c h e r s c e r t i f i c a t e i n m a t h e m a t i c s . Much o f i t would be f i t f o r a l l m a t h e m a t i c s m a j o r s a t an e a r l y s t a g e o f t h e i r u n i v e r s i t y c a r e e r .

R e f e r e n c e s on Automath:

N.G. d e B r u i j n , A s u r v e y of t h e p r o j e c t Automath.

I n : To H.B. C u r r y : E s s a y s i n c o m b i n a t o r y l o g i c , lambda c a l c u l u s and f o r m a l i s m , Academic P r e s s 1980.

L.S. v a n Benthem J u t t i n g , Checking Landau's "Grundlagen"' i n t h e A u t o n a t h s y s t e m . M a t h e m a t i c a l C e n t r e T r a c t s n r . 8 3 ,