JANUARY 19, 1912] 
Mathematicians, many philosophers, logi- 
cians and physicists, and a large number of 
other people are aware of the fact that mathe- 
matical activity, like the activity In numerous 
other fields of study and research, has been in 
large part for a century distinctively and in- 
creasingly critical. Every one has heard of a 
critical movement in mathematics and of cer- 
tain mathematicians distinguished for their 
insistence upon precision and logical cogency. 
Under the influence of the critical spirit of 
the time mathematicians, having inherited the 
traditional belief that the human mind can 
know some propositions to be true, convinced 
that mathematics may not contain any false 
propositions, and nevertheless finding that 
numerous so-called mathematical propositions 
were certainly not true, began to reexamine 
the existing body of what was called mathe- 
matics with a view to purging it of the false 
and of thus putting an end to what, rightly 
viewed, was a kind of scientific scandal. 
Their aim was truth, not the whole truth, but 
nothing but truth. And the aim was con- 
sistent with the traditional faith which they 
inherited. They believed that there were 
such things as self-evident propositions, known 
as axioms. They believed that the traditional 
logic, come down from Aristotle, was an abso- 
lutely perfect machinery for ascertaining what 
was involved in the axioms. At this stage, 
therefore, they believed that, in order that a 
given branch of mathematics should contain 
truth and nothing but truth, it was sufficient 
to find the appropriate axioms and then, by 
the engine of deductive logic, to explicate 
their meaning or content. To be sure, one 
might have trouble to “find” the axioms and 
in the matter of explication one might be an 
imperfect engineer; but by trying hard 
enough all difficulties could be surmounted 
for the axioms existed and the engine was 
perfect. But mathematicians were destined 
not to remain long in this comfortable posi- 
tion. The critical demon is a restless and 
relentless demon; and, having brought them 
thus far, it soon drove them far beyond. It 
was discovered that an axiom of a given set 
could be replaced by its contradictory and 
SCIENCE 
107 
that the consequences of the new set stood all 
the experiential tests of truth just as well as 
did the consequences of the old set, that is, 
perfectly. Thus belief in the self-evidence of 
axioms received a fatal blow. For why re- 
gard a proposition self-evident when its con- 
tradictory would work just as well? But if 
we do not know that our axioms are true, what 
about their consequences? Logic gives us 
these, but as to their being true or false, it is 
indifferent and silent. 
Thus mathematics has acquired a certain 
modesty. The critical mathematician has 
abandoned the search for truth. He no 
longer flatters himself that his propositions 
are or can be known to him or to any other 
human being to be true; and he contents him- 
self with aiming at the correct, or the con- 
sistent. The distinction is not annulled nor 
even blurred by the reflection that consist- 
eney contains immanently a kind of truth. 
He is not absolutely certain, but he believes 
profoundly that it is possible to find various 
sets of a few propositions each such that the 
propositions of each set are compatible, that 
the propositions of such a set imply other 
propositions, and that the latter can be de- 
duced from the former with certainty. That 
is to say, he believes that there are systems of 
coherent or consistent propositions, and he 
regards it his business to discover such sys- 
tems. Any such system is a branch of math- 
ematics. Any branch contains two sets of 
ideas (as subject matter, a third set of ideas 
are used but are not part of the subject mat- 
ter) and two sets of propositions (as subject 
matter, a third set being used without being 
part of the subject): a set of ideas that are 
adopted without definition and a set that are 
defined in terms of the others; a set of propo- 
sitions adopted without proof and called as- 
sumptions or principles or postulates or axi- 
oms (but not as true or as self-evident) and 
a set deduced from the former. A system of 
postulates for a given branch of mathematics 
—a variety of systems may be found for a 
same branch—is often called the foundation 
of that branch. And that is what the layman 
should think when, as occasionally happens, 
