P. HISTORY OF ' AATHEMATICAL - School of Mathematics

314 A HISTORY OF MATHEMATICAL NOTATIONS
not to be expected to be so fruitful, and, in fact, that it has not been
so fruitful.
"It is important to bear in mind that this study is avowedly concerned
with the psychological aspect of symbolism, and consequently
such a symbolism as that of Frege, which is of much less service in
the economy of thought than in the attainment of the most scrupulous
precision, is not considered.....
"In Peano's important work, dating from 1888 onwards, on logical
and mathematical symbolism, what he always aimed at, and largely
succeeded in attaining, was the accurate formulation and deduction
from explicit premises of a number of mathematical theories, such as
the calculus of vectors, arithmetic, and metrical geometry. It was,
then, consistently with this point of view that he maintained that
Rignano's criticisms hold against those who consider mathematical
logic as a science in itself, but not against those who consider it as an
instrument for solving mathematical problems which resist the
ordinary methods.
"The proper reply to Rignano seems, however, to be that, until
comparatively lately, symbolism in mathematics and the algebra of
logic had the sole aim of helping reasoning by giving a fairly thorough
analysis of reasoning and a condensed form of the analyzed reasoning,
which should, by suggesting to us analogies in familiar branches
of algebra, make mechanical the process of following the thread of
deduction; but that, on the other hand, a great part of what modern
mathematical logic does is to increase our subtlety by emphasizing
differences in concepts and reasonings instead of analogies."
699. A question.-No topic which we have discussed approaches
closer to the problem of a uniform and universal language in mathematics
than does the topic of symbolic logic. The problem of efficient
and uniform notations is perhaps the most serious one facing the
mathematical public. No group of workers has been more active in
the endeavor to find a solution of that problem than those who have
busied themselves with symbolic logic-Leibniz, Lambert, De Morgan,
Boole, C. S. Peirce, Schroder, Peano, E. H. Moore, Whitehead,
Russell. Excepting Leibniz, their mode of procedure has been in the
main individualistic. Each proposed a list of symbols, with the hope,
no doubt, that mathematicians in general would adopt them. That
expectation has not been realized. What other mode of procedure is
open for the attainment of the end which all desire?