tions. The ideal theory of polynomial domains and of algebraic function fields yields an elimination theory that is simple, conceptual, and free of formal compu- tations (HentzeP 7, E. Noether TM, v. d. Waerden19), and a systematic theory of intersection structures (Schnitt- gebilde) of algebraic manifolds (Lasker 2~ Macaulay 21, v. d. Waerden 22, Schmeidler23).

Finally, there is the extensive general module and ideal theory founded and developed by E. Noether 24. On the one hand, this theory provides a general foun- dation from which there branch off all the above arith- metic applications of ideal theory. On the other hand, it enables us to develop a comprehensive algebraic theory of general rings and their representations by matrices, including the decomposition and represen- tation theory of groups and algebras, that proceeds from a unified viewpoint and is free of computations. 25

17 Zur Theorie der Polynomideale und Resulanten (aus dem Nach- lass he_rausgegeben von E. Noether). Mathem. Ann. 88 (1923). 18 Eliminationstheorie und allgemeine Idealtheorie. Mathem. Ann. 90 (1924). i9 Neue Bergrtindung der Eliminations- und Resultantentheorie. Neiwu Archief voor Wiskunde (2) 15 (1928)~ 2~ Zur Theorie der Moduln und ldeale, Mathem. Ann. 60 (1905). 21 Algebraic Theory of Modula r Systems. Cambr idge Tracts 19 (1916). 22 Zur Nul ls te l len theor ie der Polynomideale. Mathem. Ann. 96 (1926); Eine Verallgemeinerung des. Mathem. Ann. 99 (1928). 23 Grundlagen einer Theorie der algebraischen Funktionen mehrerer Ver/inderlicher, Mathem. Zeitschr. 28 (1928). 24 Idealtheorie in Ringbereichen. Mathem. Ann. 8 (1921). 25 Hyperkomplexe Gr6f~en und Darstellungstheorie, Mathem.

As already noted, far from being limited to the clas- sical stock of algebra, the modern algebraic method goes beyond it and actually penetrates all of mathe- matics. The principle is to find the simplest conceptual foundation of a theory and in this way contribute to its unification and systematization. This principle can be applied everywhere, beginnng with logic, and, be- yond logic, in set theory, in the foundations of math- ematics, in number theory, in synthetic and analytic geometry, in topology, in the theory of integral equa- tions, and in the calculus of variations, as well as in quantum theory, which has entered algebraic waters as a result of its recent penetration by the theory of infinite matrices and abstract operators.

One must not forget that the algebraic method is just t ha t - - a method, and as such, it must be applied to something. For example, ideal theory would not have arisen for its own sake, out of interest in the definition of an ideal; it grew out of concrete problems of alge- braic number theory. The same is true of all areas of modern algebra. Thus modern algebra cannot long en- dure separation from concrete mathematical theories, and these, in turn, cannot long flourish without the systematizing and furthering influence of abstract al- gebra.

My remarks will have achieved their intended aim if they have furthered the mutual appreciation of the followers of the abstract-algebraic and concrete-math- ematical approaches and have stimulated the interest of each group in the other's methods. (Received on Sept. 16, 1929).

Comments on an essay entitled "The Modern Algebraic Method" by H. Hasse, published in 1929

Bruce Chandler and Wihelm Magnus

In the opening paragraph of his essay Hasse says: "I see it as my task to pave the way for a sympathetic understanding of modern algebra and to help its tech- niques, insofar as they are of general importance to be- come generally acknowledged . . ." It seems reason- able to examine, more than half as century later, how successful the ideas of Hasse have been. We shall try to do this not only with respect to the carefully worked out main thesis of his essay but also with regard to the asides which may represent attitudes rather than well formulated principles.

In many respects the program outlined by Hasse has been implemented so well that today's reader may be

astonished to find that it ever needed an advocate. In fact, the success of the "modern algebraic method" came rather quickly. Two years after Hasse's eassy the first (German) edition of "'Modeme Algebra" by van der Waerden appeared. There have been numerous edi- tions and translations of this textbook in the last fifty years and it is still in print. The first edition already contained the introduction of algebraic extensions ac- cording to Steinitz and the characterization of the " fundamenta l theorem of algebra" as a theorem of analysis. The second edition (1937) also contained the theory of valuations of fields and the definition of real fields in terms of non-vanishing of sums of squares,

THE MATHEMATICAL INTELLIGENCER VOL. 8, NO. 2, 1986 2 3

as well as the application of Emmy Noether's ideal theory to the theory of group representations. When the fourth edition of the English translation appeared in 1959, the epithet "Modem" was dropped from its title since, by that time, the approach was well established and even an additional chapter on topological algebra could not be considered particularly modern. Other textbooks presented much of the same material. When the authors of these texts included new material such as homology theory and universal algebra their pre- sentation of these topics was very much in agreement with the program outlined by Hasse. By the 1960's even the undergraduate curriculum had been affected: in most American colleges the s tandard theory of equations course had been replaced by an elementary course in modern algebra. Only with the increasing use and importance of computers in the 1970's does a change in attitude seem likely. It is safe to say that, as far as the teaching of algebra is concerned, Hasse's essay appears today as a statement of the obvious. But Hasse is concerned not only with teaching and not only with algebra.

What Hasse calls the "modern algebraic method" is actually the axiomatic method. The applications given belong to algebra, but the third to last paragraph--"as already noted, far from being limited to the classical stock of a l g e b r a . . . " - - s h o w s that Hasse thinks of the " m e t h o d " as re levant to all of mathematics . It is strange that he claims the method as being typical for algebra. The axiomatic method started, of course, with geometry. By 1900 the axiomatic foundation of geom- etry had been perfected, for example, in the work of Hilbert. Hilbert stated explicitly that the basic concepts do not have to be defined per se, but are adequately defined by the relations between them. Hilbert intro- duced many algebraic ideas into geometry; for in- stance, the construction and use of a non-archimedean ordered division algebra containing the real numbers. But he also "geometrized" algebra by showing that his incidence axioms together with the theorem of Pappus define a field. Clearly, Hasse was not concerned about historical facts when he wrote his essay, which ex- plains why, in his enumeration of the success of the '~modern" method, he does not mention its contribu- tion to the theory of invariants (or more precisely, polynomial invariants of the general linear group) which had been one of the most striking early suc- cesses of the method, culminating in a paper by Emmy Noether published in 1921 (Math. Annalen 83, pp. 24- 64). And "success" in this case meant not only sim- plification of the difficult proofs of Hilbert, but also "being free of formal calculations." These two aspects frequently go together, since a conceptual proof of a theorem may provide a better understanding of the theorem than a calculation. But wha t appears in Hasse's essay as result of the method, i.e., a shift away from calculation, developed in the following decades

into a dogmatic rejection of anything that looked like calculation in the textbook literature of "modern" or "abstract" algebra.

Of course, research (in contradistinction to teaching) never abol ished calculation in algebra. In the past three decades there has been a noticeable increase in the use of computation--including numerical calcu- lations using electronic c o m p u t e r s - - i n at least one field, the theory of finite groups. Here the enumera- tion of all simple groups succeeded through a joint effort of mathematicians using every available tool. We certainly hope that someday a general principle will be found to deduce the existence and structure of ex- actly 26 "sporadic" simple groups. We also hope that such a principle will not obliterate the connection be- tween these groups and the denses t packing of spheres in 24-dimensional space and that it will even shed light on the mysterious and highly intriguing dis- covery by D. O. McKay of the relation between the degree of the irreducible representations of the largest groups and the coefficients in the power series of an elliptic modular function.

The concluding remarks about "concrete mathemat- ical theories" in Hasse's essay show that he did not reject calculation per se and this can be confirmed by inspecting the textbook he wrote. Similarly, Hasse himself cannot be charged with an act of omission which, according to C. L. Siegel, has been committed by the adheren t s of the " m o d e r n " or "abs t rac t " method in algebra. In a paper on Bernoulli polyno- mials and quadratic fields (Gesammelte Abhandlungen, Vol. 4, 1979) Siegel, on p. 33, offers a sarcastic apology for giving examples. He writes:

T o d a y . . . one is satisfied with the statement that the latest just discovered generalization can easily be confirmed in the case of a field with two elements or a group of one element or a set of zero elements.

It would be impossible to support Siegel's appraisal with statistics. However, it is rather obvious that ex- amples of a non-trivial type involve special arguments as well as calculations. For example, the old algebra textbook by R. Fricke (1925) contains no exercises, and uses, by today 's standard, outdated and inelegant methods of proof. But it contains an incredible wealth of highly non-trivial examples.

Most of our comments, as well as Hasse's, refer to algebra. However, Hasse mentions other areas, espe- cially analysis. Here his att i tude appears to change during the course of his essay. In the first paragraph, Hasse refers to " . . . analysts whose approach and methods are directly opposi te to modern algebraic techniques." Later, he mentions that in analysis one refers to the function of measurement as something important for applications and also says that refer- ences to intuitive appeal and applicability are foreign to the algebraic method. Nevertheless, at the end of

24 THE MATHEMATICAL INTELLIGENCER VOL. 8, NO. 2, 1986

the essay, Hasse mentions integral equations, the cal- culus of variations, and even quantum theory as fields which are open to the algebraic method.

Hasse could not have anticipated the important paper by J. von Neumann on "Eigenwerttheorie her- mitescher Funktionaloperatoren" (Math. Ann. 102, 1930, p. 49-131). This paper definitely does not fit in with Hasse's general remarks on analysis at the begin- ning of his essay, although it would support his later comments on the range of the algebraic method. As a matter of fact, von Neumann's paper documents the merits of the abstract method by showing that certain results in operator theory cannot be obtained by using the older "concrete" method involving infinite ma- trices. Since then, the theories of distr ibutions, of Banach spaces, and of other abstract concepts in anal- ysis definitely render obsolete Hasse's remarks con- trasting analysis to modern algebra. Even the disdain for computations is found among analysts; we have heard a leading analyst say something about "those obnoxious explicit solutions of boundary value prob- lems."

Probably the most important aspect of Hasse's essay is the author's interest in the special status of the ar- chimedean valuation or, equivalently, in the problem of "equal rights" for the archimedean and the non- archimedean valuations of the rational numbers and their topological completions. Since Hasse wants to find out whether the archimedean valuation can claim to be of special interest from the point of view of modern algebra, he disregards its importance in anal- ysis and also the fact that our knowledge of the phys- ical universe is based on an archimedean valuation. Under these circumstances, Hasse 's idea of finding

how far analytic continuation can be carried out for functions defined by power series based on a non- archimedean valuation recommends itself as an un- assailable test. The paper by his student W. Schbbe, which Hasse announces in his essay, explores this idea. We have not been able to obtain the paper (it appeared in an obscure place: Universit~itsarchive 42, Abteilung No. 2, 63 pages) but a very informative re- view of it is in the Jahrbuch ~iber die Fortschritte der Math- ematik in 1930. According to this review, Sch6be showed that in the case of a p-adic valuation analytic continuation can be achieved only through reordering ("Umbildung") of power series, and that this is pos- sible only if a power series expansion at the point at infinity exists. For the large number of other results we refer to the review or the original paper.

Hasse was a number theorist and in his work he was, like most mathemat ic ians , a pragmatis t who would use any available tool without asking method- ological questions, although he might have preferred purely algebraic proofs to those using geometry or analysis. However, his essay is curiously dogmatic. Even if we consider only algebraic statements, would it be desirable to eliminate Minkowski's geometric ar- gument from the proof that every algebraic extension of the field or rational numbers is ramified? And as for the special posi t ion of the a rch imedean valuation: Could one not simply observe that it is the conse- quence of the natural ordering of the integers which, in turn, is the basis of complete induction, a tool par- ticularly indispensable for the number theorist? It is also ironic that two years after Hasse's essay GbdeP 6 published his incompleteness proof showing the lim- itations of the axiomatic method.

Mathematics and the External World Continued from page 16

with possible appl icat ions of statistical-empirical methods in analyzing historical texts. Several new methods I have proposed for dating texts sometimes make it possible to establish a more precise chro- nology of ancient events. The results do not neces- sarily agree with traditional chronology.

However, this is a complicated and controversial subject, in which much remains to be done before full clarity is achieved. For this reason I would have to give explanations in much greater detail than is appro- priate for this interview in order to describe examples of possible redating of ancient events. So I will have to refer the interested reader to my publications on the subject for such details. There is only one easily acces- sible publication in English: my article "Global Chro- nology and New Methods of Dating Ancient Events"

will soon appear in English translation.

Q. Do you see an increasing use of mathematics in humanitarian disciplines such as history? What are the possibilities and limitations in the use of mathematics in those areas?

A. In m y o p i n i o n , h i s t o r y is on the th re sho ld of a major i n t r oduc t i on of stat ist ical methods. This development seems to me to be useful both for history and for mathematics. Of course, mathematical methods must play an auxiliary role, by enabling one to analyze large quantities of information and formulate various statistical conjectures. Such conjectures must ultimately be verified by historical specialists (in cooperation with scientists).

Dept. of Mathematics GN-50 Univesity of Washington Seattle, Washington 98195

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