722 REPORT—1890. 
the sake of its applications to the concrete sciences, it must indeed seem that it has 
long since run wild, and expanded itself into a thousand useless extravagances. 
Eyen the mathematician must sometimes ask himself the questions—not un- 
frequently put to him by his friends—‘To what is it all tending? What will be 
the result of it all? Will there be any end?’ The last question is readily 
answered, There certainly can be no end ; so wide and so various are the subjects 
of investigation, so interesting and fascinating the results, so wonderful the fields of 
research laid open at each succeeding advance—no matter in what direction—that 
we may be sure that, while the love of learning and knowledge continues to exist in 
the human mind, there can be no relaxation of our efforts to penetrate still further 
into the mysterious worlds of abstract truth which lie so temptingly spread 
before us. The more that is accomplished, the more we see remaining to be 
done. Every real advance, every great discovery, suggests new fields of inquiry, 
displays new paths and highways, gives us new glimpses of distant scenery. 
This wonderful suggestiveness is itself one of the marks of a true theory, one 
of the signs by which we know that we are investigating the actual, exist- 
ing truths of nature, and that our symbols and formule are expressing facts 
quite independent of themselves, though decipherable only by their means. As 
for the other questions, it is very difficult to render intelligible even to a 
mathematician the kind of knowledge acquired by mathematical research in a new 
field until he has made himself acquainted with its processes and notation, and we 
cannot hope to find in the remote regions of an abstract science many results so 
simple and striking as to appeal forcibly to the imagination of those who are 
unfamiliar with its conceptions and ideas. It would seem therefore that the 
question, ‘To what is it all tending P’ could never be answered in general terms. 
I do not think any mathematician could see his way to a reply, or even 
give definite meaning to the question. He might feel daring enough to predict 
the probable drift of his own subject, but he could scarcely get a broad enough 
view to enable him to indulge his fancy with respect to more than a very 
minute portion of the field already open to mathematical investigation. To the 
outsider I am afraid that the subject will continue to present much the same 
appearance as it does now; it will always seem to be stretching out into limitless. 
symbolic wastes, without producing any results at all commensurate with its 
expansion, 
Instead of attempting to consider the general question of what may be expected 
to result from the progress of mathematical science, we may restrict ourselves to: 
asking whether the great extension of the bounds of the subject which is taking 
place in our time, will materially add to its powers as a weapon of research in the 
concrete subjects. This is a question of the highest interest, and one that cannot 
fail to have occupied the thoughts of every mathematician at some time or another 
in the course of his work. For my own part, I do not think that the bearing of 
the modern developments of mathematics upon the physical sciences is likely to be 
very direct or immediate. It would indeed be rash to assert that there is any 
branch of mathematics so abstract or so recondite that it might not at any moment 
find an application in some concrete subject ; still, it seems to me that, if the exten- 
sion of the pure sciences could only be justified by the value of their applications, 
it is very doubtful whether a satisfactory plea for any further developments could 
be sustained. As a rule each subject involves its own ideas and its own special 
analysis, and it can only occasionally happen that analytical methods devised for 
the expression and development of one subject will be found to be appropriate for 
another. It is obvious also that the chance of such applications becomes less and 
less as we travel farther and farther from the elementary processes and methods 
which are common to all the exact sciences. There is a general resemblance of 
style running through much of the analysis required in the physical sciences, but 
there is no such resemblance in the case of the pure sciences, or between the pure 
and the physical sciences. It appears likely therefore that, in the future, the 
mathematical obstacles which present themselves in physical research will have to 
be overcome, as heretofore, by means of investigations undertaken for the purpose, 
Cee 1.) ee 
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and that analysis will continue to be enriched by conceptions and results, and . ; 
n 
