sign in sign up
<<
Home , David Hilbert

HISTORICAL NOTES

The most influential speech ever presented in the history of

Rosalind Ezhil K.

In the year 1897, the very first ‘Interna- 5. Lie’s concept of a continuous group either the quantitative or in the qualita- tional Congress of Mathematicians’ (ICM) of transformations without the assump- tive regard’2. However, the importance was convened in the city of Zurich. It tion of the differentiability of the func- of his talk was recognized rather quickly was a time when Europe was at the fore- tions defining the group. afterwards. The prediction of his close front of mathematical study and the Con- 6. Mathematical treatment of the axi- friend and fellow mathematician Hermann gress was organized by leading French oms of . Minkowski ‘Most alluring would be the and German mathematicians of the day. 7. Irrationality and transcendence of attempt to look into the future, in other It was decided at that meeting that the certain numbers. words, a characterization of the problems Congress would be a regular in the 8. Problems of prime numbers. to which the mathematicians should turn coming years with one of the primary 9. Proof of the most general law of re- in the future. With this, you might con- purposes being ‘to promote personal ciprocity in any number field. ceivably have people talking about your relations between mathematicians of dif- 10. Determination of the solvability of speech even decades from now’3 was to ferent lands. . .’ Since then, except for a . come true many times over. the period corresponding to the two 11. Quadratic forms with any algebraic In fact, the impact of his talk on the World Wars, the Congress has been a numerical coefficients. course of mathematical research in the rather regular event. The next Congress 12. Extension of Kronecker’s 20th century was extraordinary. Cer- is due to happen in 2010 (ICM 2010) and on abelian fields to any algebraic realm tainly none of the other ICMs that took for the first time in its history, the Con- or rationality. place after the 1900 can boast of an event gress will take place in India. 13. Impossiblity of the solution of the with consequences of such magnitude. In this article, we will look at a parti- general equation of the 7th degree by Solving one of ‘Hilbert’s problems’ is a cularly significant event that took place means of functions of only two arguments. matter of great prestige and has been the in the second Congress ever to be held. 14. Proof of the finiteness of certain preoccupation and inspiration of a great This Congress was held at in 1900. complete systems of functions. number of mathematical minds. A large At this meeting, leading German mathe- 15. Rigorous foundation of Schubert’s number of articles, papers, books have matician of the day – David Hilbert, was enumerative calculus. been written on the subject. invited to present a talk. And on the 16. Problem of the topology of alge- What is the status of the problems morning of 8 August 1900, during the braic curves and surfaces. today? Some of the problems that Hilbert section of the Congress on bibliography 17. Expression of definite forms by presented were solved fairly quickly; and history, David Hilbert presented squares. some were decided not to be problems at his talk entitled, ‘The future problems of 18. Building up of space from congru- all whereas some continue to remain mathematics’. He outlined some 23 pro- ent polyhedra. open to this day. In contrast to mathema- blems in different fields of mathe- 19. Are the solutions of regular pro- ticians of today, Hilbert was a ‘universal’ matics that he believed were a sample of blems in the always mathematician with a profound under- what future mathematicians ought to be necessarily analytic? standing of many unrelated areas of working at. (He had time during the 20. The general problem of boundary mathematicians. As Ben Yandell4 says in lecture to present only ten problems – the values. his book (The Honors Class: Hilbert’s rest were part of the published proceed- 21. Proof of the existence of linear dif- Problems and their Solvers): ‘Mathemat- ings.) ferential equations having a prescribed ics has come a long way. . . , expanding It would be a daunting task to explain monodromic group. and fragmenting to the that it is in- these problems even to a sophisticated 22. Uniformization of analytic rela- conceivable that one person could survey mathematical audience – we will there- tions by means of automorphic functions. the whole and make a similar list of fore not even attempt it here! However, 23. Further development of the meth- problems. In the worst case such a list to give a flavour of the talk, we present ods of the calculus of variations. would be the work of a series of commit- the list of the 23 problems here. tees with no common members. The Did this talk have an instant impact? International Mathematical Union did 1. Cantor’s problem of the cardinal Apparently, British mathematician Char- better than this in 2000, when it pub- number of the continuum. lotte Angas Scott who prepared a report lished Mathematics: Frontiers and Per- 2. The compatibility of the arithmeti- on the proceedings for the Bulletin of the spectives (“It is inspired by the famous cal . American Mathematical Society has re- list of problems that Hilbert proposed”) 3. The equality of the volumes of two corded only ‘a rather desultory discus- but still credited four editors, had tetrahedra of equal bases and equal alti- sion’2 after the talk. And Hilbert himself thirty signed articles, and was 459 pages tudes. was not very pleased with the confer- long’. 4. Problem of the straight as the ence – he wrote to his friend Adolf Hur- Born in 1864 in the town of Konisberg shortest distance between two points. witz that ‘the visit was not very strong in (then a part of the German Kingdom of

CURRENT SCIENCE, VOL. 97, NO. 10, 25 NOVEMBER 2009 1493 HISTORICAL NOTES

Prussia, now a part of Russia), Hilbert way’2 from the viewpoint Poincaré put fields of our science. Trying to unveil showed very early on, his extraordinary forward. what the future would hold in store for flair and passion for mathematics. At the We end with a tribute to David Hilbert us, he posed and discussed 23 unsolved time of his entry into the world of mathe- in the words of his student Hermann problems which have indeed, as we can matics, the tenet of French philosopher Weyl, also a great mathematician, ‘A now state in retrospect, played an impor- Du Bois-Reymond that some scientific great master of mathematics passed away tant role during the following 40 odd problems were unsolvable was much in when David Hilbert died in Göttingen on years. A mathematician who had solved vogue. However, this was something February the 14th, 1943, at the age of one of them thereby passed on to the Hilbert completely rejected and fought eighty-one. In retrospect it seems to us honours class of the mathematical com- against all his life. that the era of mathematics upon which munity6.

At the age of 38 when he presented his he impressed the seal of his spirit and bold talk, Hilbert ended his famous lecture which is now sinking below the horizon 1. Wikipedia. thus ‘There are absolutely no unsolvable achieved a more perfect balance than 2. Ivor Grattan-Guinness, Notices of the Am. problems. Instead of the foolish ignor- prevailed before and after, between the Math. Soc., 2000, 47, 752. abimus, our answer is on the contrary: mastering of single concrete problems 3. Jeremy, G., Newsletter 36 of the European We must know, we shall know’. A ‘pure’ and the formation of general abstract Mathematical Society, 2000, pp. 10–13. 4. Benjamin H. Yandell, The Honors Class: mathematician at that stage in his life, concepts. Hilbert’s own work contributed Hilbert’s Problems and Their Solvers. Hilbert was not very interested in the ap- not a little to bringing about this happy 5. Sridharan, R., Resonance, 1999, 4, 96. plications of mathematics. His talk is equilibrium, and the direction in which 6. Herman Weyl, Bull. Am. Math. Soc., 1944, considered to be a response to mathema- we have since proceeded can in many in- 50, 612–654. tician Poincaré’s speech on the impor- stances be traced back to his impulses. tance of applied mathematics at the No mathematician of equal stature has Rosalind Ezhil K. lives at 122, 3rd Main, previous ICM in Zurich. In fact there is risen from our generation. . .’. Dollars Colony, RMV 2nd Stage, 1st some criticism of his talk as having His Paris address on ‘Mathematical Block, Bangalore 560 094, India. ‘. . .surely swung too much the other problems’ quoted above straddles all e-mail: [email protected]

Edited and published by P. Balaram, Current Science Association, Bangalore 560 080. Typeset by WINTECS Typesetters (Ph: 2332 7311), Bangalore 560 021 and Printed at Printek Printers, Bangalore (Ph: 2328 7763)

1494 CURRENT SCIENCE, VOL. 97, NO. 10, 25 NOVEMBER 2009