MODERN MATHEMATICAL RESEARCH MILLER. 189 



reply that you did not see how you coukl marry now, as you had not 

 yet become acquainted with all the young ladies ?" 



In connection with this remark by Jacobi we may recall a remark 

 by another prominent German mathematician who also compared 

 the choice of a subject of research with marriage. In the "Fest- 

 sclu'ift zur Feier des 100 Geburstages Eduard Kummer," 1910, page 

 17, Prof. Hensel states that Kummer declined, as a matter of prin- 

 ciple, to assign to students a subject for a doctor's thesis, saying 

 that this would seem as if a young man would ask him to recom- 

 mend a pretty young lady whom this young man should marry. 



While it may not be profitable to follow these analogies into 

 details, it should be stated that the extent to which a subject has 

 been developed does not necessarily affect adversely its desirability 

 as a field of research. The greater the extent of the development 

 the more frontier regions will become exposed. The main question 

 is whether the new regions which lie just beyond the frontier are 

 fertile or barren. This question is much more important than the 

 one which relates to the distance that must be traveled to reach 

 these new fields. Moreover, it should be remembered that mathe- 

 matics is n-dimensional, n bemg an arbitrary positive integer, and 

 hence she is not limited in her progress to the directions suggested 

 by our experiences. 



If we agree with ]\Iinkowski that the integers are the source of all 

 mathematics,^ we should remember that the numbers which have 

 gained a ])lace among the integers of the mathematician have in- 

 creased wonderfully during recent times. According to the views 

 of the people who preceded Gauss, and according to the elementary 

 mathematics of the present day, the integers may be represented 

 by ])oints situated on a straight line and separated by definite fixed 

 distance. On the other hand, the modern mathematician does not 

 only fdl up the straight line Avith algebraic integers, placing them 

 so closely together that between any two of them there is another, 

 but he fills up the whole plane equally closely with these mtegers. 

 If our knowledge of mathematics had increased during the last two 

 centuries as gi'eatly as the number of integers of the mathematician 

 we should be much beyond our present stage. The astronomers 

 may be led to the conclusion that the universe is probably finite, 

 from the study of the number of stars revealed by telescopes of 

 various powers, but the mathematician finds nothing which seems to 

 contradict the view that his sphere of action is infinite. 



From what precedes one would expect that the number of fields 

 of mathematical research appears unlimited, and this may serve to 



1 This view was expressed earlier by Kroneeker, who was the main founder of the school of mathemati- 

 cians who aim to make the concept of the positive integers the only foundation of mathematics. Cf. Klein 

 und Schimmack, " Der mathematische Unterricht an deu hohercn Schulen," 1907, p. 175. 



