PROBLEMS EMPHASIZED BY PR A GMATISM 40 5 



forms simply does not consist in an analysis of the premises. Eor is 

 it intended to make yon believe the conclusion. The true interest of 

 deduction lies in the fact (1) that it is a process possessing an in- 

 exhaustible fecundity, and (2) that this fruitfulness results in giving 

 you a knowledge that certain premises imply certain conclusions. 



IV 



But now let me briefly put before you some genuine deductive 

 processes, and point out what problems with regard to their psychology 

 can be aroused. Let me begin with the instance with which I have often 

 wearied my friends, so that some of you who are here present will have 

 heard me mention it. Eaise the question whether a strip of paper can 

 exist which shall have only one side. One would be disposed to settle 

 the question empirically by observing that every strip of paper in one's 

 ordinary experience obviously has two sides. If one passes from a con- 

 sideration of strips of paper that have two ends, to the consideration 

 of endless strips of paper, that is strips of paper made in ring shape, 

 one sees that in an ordinary ring the two-sidedness of strips of paper 

 still holds good. But if one takes an ordinary strip of paper, say two 

 inches wide and eight or ten inches long, first twists one end of it 

 180 degrees, and then brings the two ends together, one has the result- 

 ing geometrical form of the one-sided ring. The first discovery that 

 such a ring is possible was of course an empirical discovery. But the 

 geometers (I believe it was Mobius who first noticed one-sided surfaces) 

 had their attention at once attracted to the mathematically interesting 

 properties of this form. Now when such a form is viewed as a mathe- 

 matical object, any one with mathematical interest naturally proceeds 

 to an undertaking of the sort which is characteristic of mathematical 

 science in general. One endeavors to deduce some of these properties 

 from others, or to discover, as the ordinary mind would say, why these 

 properties belong to any one-sided surface. Hereupon let me mention a 

 problem that can be studied as soon as you have once taken note of the 

 one-sided surface and have begun to make a study of the real sense or 

 connection of its structure. Suppose a one-sided surface, a ring strip 

 of the sort that I have described, to be cut down the middle, midway 

 between what appear to be the two edges of the strip, and suppose the 

 cut continued until it returns into itself, what will be the result? 

 There are two ways of answering the question. One is the directly 

 empirical one of making the cut. The other method is to endeavor to 

 see before making the strip what must be the result in view of the one- 

 sidedness of the strip. I once proposed the question to a class, and found 

 a member of the class, who although not a student of mathematics, pos- 

 sessed a relatively clear visual imagination, was ingenious, became inter- 

 ested in the problem, solved it without cutting the strip, then tested 



