subconscious mandate to announce the end of all “rational” or 
visual values. He was acting like Pip in Dickens’ Great  
Expectations . Pip was a poor boy who had a hidden benefactor 
who wanted to raise Pip to the status of a gentleman. Pip was 
ready and willing until he found that his benefactor was an 
escaped convict. Spengler and Hitler and many more of the 
would-be “irrationalists” of our century are like singing-telegram 
delivery boys, who are quite innocent of any understanding of 
the medium that prompts the song they sing.

     So far as Tobias Dantzig is concerned in his Number: The  
Language of Science , the progress from the tactile fingering of 
toes and fingers to “the homogeneous number concept, which 
made mathematics possible” is the result of visual abstraction 
from the operation of tactile manipulation. We have both 
extremes of this process in our daily speech. The gangster term