ON THE UNCERTAINTY OF CONCLUSIONS: ^ !' 5 
duties upon imported merchandise. All have access to the 
same store of experience ; the discussions and investigations 
of the past are open to all alike. In the end, however, their 
conclusions, even as to elementary principles, are diamet¬ 
rically opposed to each other. 
But I have neither the time nor the disposition to enter 
into an exhaustive examination into the miscarriage of logic 
in the regions of politics, religion, or social science. I must 
restrict myself to some consideration of the uncertainty of 
conclusions reached by what may be broadly included under 
the general term “ the exact sciences,” a division of the sub¬ 
ject not unlikely, I hope, to be of some interest to members 
of this Society. 
At the threshold of the investigation we are confronted by 
the term “ exact sciences ,” and it is of the utmost importance 
to reach a clear understanding of the meaning of this phrase 
in the beginning. By some writers its application is limited 
to the mathematical sciences or substantially to pure math¬ 
ematics. This does not seem, however, to be in accord with 
the general usage among scientific men, and a wider signifi¬ 
cance will be here given to it. 
Pure mathematics may, and posssibly must, be regarded 
as a mode of thought; as symbolic logic; as an abridgment 
of mental processes by the selection of that which is com¬ 
mon to all, and its formal expression by means of signs and 
symbols. Intellectual operations which, on account of their 
complexity and length, would be possible only to a few of 
the highest capacity are by the aid of mathematics brought 
within the range of the many. In virtue of the simple and 
beautiful nomenclature of the science, one can see at a glance, 
in a formula or equation, the various relations, primary and 
secondary, direct and implied, which exist among the sev¬ 
eral magnitudes involved, which, if expressed or defined in 
ordinary language, would be beyond the understanding of 
most intelligent people. 
The principles and rules governing mathematical opera¬ 
tions have been, in the main, so well worked out and so uni- 
