62 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
artificial individual spring the related ideas “one” and “many.” A 
unit thought of in contrast to “ many” as not-many gives us the idea 
one or “ a one.” A “ many” composed of “a one” and another “ one” 
is characterized as “ two.” A many composed of “ a one” and the spe¬ 
cial many “a two” is characterized as “three.” And so on, at first 
absolutely without counting, in fact before the invention of that patent 
process of identification now called counting. For a considerable period 
of its early life every child uses a number system consisting of only three 
terms, one, two, many, and no counting. The undue importance given 
in this system to the number two is fossilized in the duals of Greek gram¬ 
mar, and indicated by the unnecessary richness of languages in such 
terms as a pair, a couple, a yoke, a span, a brace, etc. 
The “Anzahl” of a group is wholly abstract, in that it represents all 
at once the primitive individuals or elements of the group or artificial 
individual, and nothing more. There never was and never will be a con¬ 
crete number or anything concrete about number. 
The number in the sense of “Anzahl” of a group is a selective photo¬ 
graph of the group, a numeric picture which takes or represents only one 
quality of the group, but takes that all at once. This picture process 
only applies primarily to those particular artificial wholes which may be 
called discrete aggregates. But these are of inestimable importance for 
human life. 
This overwhelming importance of the number-picture after centuries 
led to a human invention as clearly a device of man for himself as is the 
telephone. This was a device for making a primitive individual think¬ 
able as a recognizable and recoverable artificial individual of the kind 
having the numeric quality. This is the recondite device called meas¬ 
urement. 
Measurement is an artifice for making a primitive individual conceiv¬ 
able as an artificial individual of the group kind, and so having an 
“Anzahl,” a number picture. 
It may be likened to dyeing cotton with analine dyes. This will give 
the cotton a color which may then be identified by comparison with the 
set of standard colors. 
The height of a horse, by use of the artificial unit, a “ hand,” is think¬ 
able as a discrete aggregate and so has a number-picture identifiable by 
comparison with the standard set of pictures, that is by counting, as say 
16. But to argue from this the implicit presence of the measurement 
idea in every number is the analogue of maintaining the implicit pres¬ 
ence of the process-of-dyeing idea in every color. 
Remark on the preceding by Dr. Edmund Montgomery: “ I think you 
are right in conceiving the ‘ essence of number’ to consist in the fact 
