JUNE 7, 1912] 
also reply that you did not see how you 
could marry now, as you had not yet be- 
come acquainted with all the young ladies? 
In connection with this remark by Jacobi 
we may recall a remark by another promi- 
ment German mathematician who also com- 
pared the choice of a subject of research 
with marriage. In the ‘‘Festschrift zur 
Feier des 100 Geburtstages Eduard Kum- 
mers,’’ 1910, page 17, Professor Hensel 
states that Kummer declined, as a matter 
of principle, to assign to students a subject 
for a doctor’s thesis, saying that this would 
seem to him as if a young man would ask 
him to recommend to him a pretty young 
lady whom he 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 
mathematics is n-dimensional, n being an 
arbitrary positive integer, and hence she is 
not limited, in her progress, to the direc- 
tions suggested by our experiences. 
If we agree with Minkowski that the 
integers are the source of all mathematics? 
we should remember that the numbers 
which have gained a place among the in- 
tegers of the mathematician have increased 
' This view was expressed earlier by Kronecker, 
who was the main founder of the school of mathe- 
maticians who aim to make the concept of the 
positive integers the only foundation of mathe- 
maties. Cf. Klein und Schimmack, ‘‘Der mathe- 
matische Unterricht an den hoeheren Schulen,’’ 
1907, p. 175. 
SCIENCE 
879 
wonderfully during recent times. Accord- 
ing to the views of the people who preceded 
Gauss, and according to the elementary 
mathematics of the present day, the in- 
tegers may be represented by points situ- 
ated on a straight line and separated by 
definite fixed distance. On the other hand, 
the modern mathematician does not only 
fill up the straight line with algebraic in- 
tegers, placing them so closely together 
that between any two of them there is an- 
other, but he fills up the whole plane 
equally closely with these integers. If our 
knowledge of mathematics had increased 
during the last two centuries as greatly as 
the number of integers of the mathema- 
tician we should be much beyond our pres- 
ent 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 noth- 
ing 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 furnish a partial explanation of 
the fact that it seems impossible to give a 
complete definition of the term mathe- 
matics. If the above view is correct we 
have no reason to expect that a complete 
definition of this term will ever be possible, 
although it seems possible that a satisfac- 
tory definition of the developed parts may 
be forthcoming.® 
Among the various fields of research 
those which surround a standing problem 
are perhaps most suitable for a popular 
exposition, but it should not be inferred 
that these are necessarily the most impor- 
tant points of attack for the young inves- 
°Boécher discussed some of the proposed defini- 
tions in the Bulletin of the American Mathematical 
Society, Vol. II. (1904), p. 115. 
