Algorithms Are Not EnoughAuthor(s): Peter HiltonSource: The College Mathematics Journal, Vol. 16, No. 1 (Jan., 1985), pp. 8-9Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2686618 .

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arrive, like the theory of algorithmic complexity, which deal, to my taste, with the most fascinating and difficult open questions of contemporary mathematics (like "P = /VP?"). This accompanies a further development of discrete mathematics. Classical continuous methods can not be used to tackle these problems. Over?

simplified, we can state that algorithmic and discrete mathematics have a strong mutual influence and are, to a certain extent, synonymous. Even P. Halmos, who is

certainly not a protagonist of discrete, algorithmic or applied mathematics, has to admit (p. 19 in L. A. Steen, ed., Mathematics Tomorrow, Springer, 1978) "... that in the foreseeable future (as in the present) discrete mathematics will be an

increasingly useful tool in the attempt to understand the world and that analysis will therefore play a proportionally smaller role ... ."

There is no doubt, we have to go the algorithmic way. It does not allow an easy ride in a well-known neighborhood with traffic lights and street signs in a powerful car with automatic gear shift. It is a demanding hike into the backcountry of an unknown territory with risk and danger, but with the possible rewards of beautiful and new knowledge and of fascination.

Good luck to all who join to take this hike!

Algorithms are Not Enough . . . Peter Hilton, SUNY at Binghamton, Binghamton, NY

It is always tempting to believe one has discovered a panacea. Stephen Maurer has fallen victim to this temptation. Students have, distressingly

often, been unsuccessful at learning and understanding mathematics. Computers have brought algorithms to the forefront and sparked new interest in algorithmic proofs. So let us solve our pedagogical problems by concentrating on algorithmic arguments. This, basically, is his message.

Certainly, algorithmic arguments are important, powerful and?often? attractive. But it would be disastrous to adopt them exclusively. Consider the

following argument. Let a,b be any two non-zero integers. Consider all linear combinations ka + //?, where /:,/ are integers. There will be a smallest positive integer d in this set. Using the Euclidean property of Z, one quickly proves that d = gcd(a,b); so one has proved that gcd(a,b) is a linear combination of a and b. This can be proved algorithmically, of course, but such a proof suffers both in

efficiency and esthetically. However, if one replaces {a,b} by an infinite set of

integers, it is not clear that an algorithmic approach is even feasible, whereas the

given proof remains essentially the same. Let us consider one other example. There is a very nice non-algorithmic proof

that the multiplicative group of non-zero residues modulo a prime p is cyclic?an important result in elementary number theory. How could one prove this algorith? mically? We would presumably have to have a means of knowing which are the

primitive residues mod/? (the generators of this cyclic group); but we still don't even know if 2 is primitive for infinitely many p.

There are, in fact, many reasons for not paying exclusive attention to algorithmic arguments. Let us offer a few of the most important to supplement those already given.

There is no algorithmic way to design algorithms. We should want students to find their own proofs. There is no algorithmic way to do this. It is common ground that students will often understand proofs better if these proofs consist of actually

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finding the 'solution' in given (but typical) numerical cases. But we want students to think about mathematical situations, not simply to have available a catalogue of

recipes. They should be encouraged to invent proofs, at least in relatively simple situations.

Human beings are not machines. Computers execute algorithms superbly; that is their mode. We should not be training our students to imitate machines. Of course, that is not what Maurer is advocating, but undue concentration on algorithms would eliminate from the mathematics with which the student becomes familiar those parts which are uniquely human.

Algorithms disguise generality. Maurer himself gives the example of the axiomatic

approach to the matrix equation Ax = b compared to Gaussian elimination. But the axiomatic approach tells you many things Gaussian elimination doesn't. It tells you about solving systems of linear differential equations. It tells you about the

counterimage of an element of B under a group homomorphism <J>: A -? B. It also tells you what to look for in a particular numerical case, and is therefore slighted by Maurer's stricture that it 'isn't any help.' The axiomatic approach establishes links between different parts of mathematics and these are important, even critical. Should we convince ourselves that 712 = 1 (mod 13) by multiplying up on our hand-calculator or by understanding Fermat's Theorem?

Finally, let me say something about infinite sets. Maurer admits that you 'can't

prove things about infinite sets with algorithms,' although he seems to withdraw the admission immediately. But it is a subtle matter to specify what 'things about infinite sets' can be treated by algorithms. We can certainly use an algorithm to determine the smallest element in an infinite set of positive integers?even a machine could do this quickly if the set were ordered. Surely Maurer must be

talking about infinite sets when he quotes Hermann Weyl, since Weyl was certainly thinking of such theorems as the Brouwer Fixed Point Theorem. The issue Maurer raises concerns more than a choice of proofs, however; it is ultimately concerned with a choice of mathematics. That the computer should be influencing the curriculum has been argued persuasively by Maurer and others [see The Future of College Mathe? matics, edited by Anthony Ralston and Gail S. Young, Springer-Verlag, 1983]; but it is not common ground that it should dominate the curriculum. Certainly we should include far more numerical work in our teaching of algebra and calculus; but it has been demonstrated by so many examples that computer availability does not remove the need for mathematical analysis. And if mathematical analysis of a

problem is important, then we must teach students how to do it. The learning and memorization of algorithms will not suffice. An appreciation of the significance of

algorithms, however, is not only important in itself, but can also enhance the students' mastery of mathematical analysis.

The Path to Hell . . . Peter Renz, Bard College, Annandale on Hudson, NY

A pavement of good intentions gives no guarantee of reaching paradise. We should aim to make the mathematics we teach more algorithmic, but the way in which this is done must be worked out more carefully than in some parts of Maurer's position paper. To indicate the direction such a debate might take, and the dangers of

oversimplifying, I will look more clearly at some of his specific suggestions. Some algorithms are not what they seem. It may be comforting for a student to

think that the integral of f on [a,b] can be given by a simple algorithmic formula such as

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