INTRODUCTION TO THE SECOND EDITION lv 

The old orientations are being perpetuated, as a rule unknowingly, through the aristotelian structure of our language, our institutions, etc. The new orientations are simpler than the old because they are closer to empirical facts, and so are even more easily absorbed by children  provided parents, teachers, etc., are themselves aware of the new methods and so can give the children guidance.
The difficulties we are now facing, with the many important new factors introduced in a nonaristotelian system, listed roughly in the tabulation, cannot be evaluated effectively unless we understand the role that new factors play in our generalizations.
Section E. New factors: the havoc they play with our generalisations.
In mathematics and science we use extensively the method of interpolation. In building curves we do not have all the points or data. We have a number of them and then connect the points with a smooth curve. 



The equation of that curve is given on the basis of the actual data at hand. The nervous processes which are involved in interpolations and building up equations are also involved in producing ordinary generalisations in daily life; that is, we interpolate from the data we have and then generalize in words instead of equations. It is well known that sometimes when a new datum is discovered it transforms the curve entirely, with a corresponding change in the equation (generalization).
Fig. 2 as an illustration will make this clearer. If we measure the experimental points (1,0), (3,6), (5,12), we would find them to lie on the line abc with the equation y=3x3, and we might conclude therefrom that further similar experiments would confirm the linearity of the relationship being studied. But if a further analysis yields the point (2,6), the simplest curve fitting these data is now the curve adbec, expressed by the equation y = x^{3}  9x^{2} + 26x 18, which is different and much more complex than before, because it is a cubic equation instead of a linear equation.*


♦I am indebted for this example to Dr. A. S. Householder, University of Chicago. 
