246 INTRODUCTION TO PHILOSOPHY OF SCIENCE 



variability in the levels of abstraction at which the scientific 

 concepts may be analyzed. For example, time and space on 

 the level of the Newtonian physics are not the same as the 

 time and space of relativity; and number may be defined 

 in such a way as to include only the positive cardinal num- 

 bers, or it may be generalized successively so as to include 

 the negatives, the fractions, the irrationals, and the imag- 

 inaries. The situation is even more confusing than this, for 

 by time and space may be meant simply the measured 

 values obtained by reading clocks and scales, and by num- 

 ber may be meant simply the rules according to which 

 certain arbitrarily chosen marks may be combined and 

 substituted for one another. At these extreme levels it seems 

 safe to say that the scientific concept has more or less com- 

 pletely lost its content; by attaining extreme generality of 

 application it has lost the features which distinguish it 

 from any other concept. The movement of science seems 

 to be always toward greater and greater abstraction. For 

 example, the attempt is made to identify the concepts of 

 mathematics with purely logical notions, the concepts of 

 mechanics with the concepts of mathematics, the concepts 

 of physics with the concepts of mechanics, and so on. Hence 

 one must recognize the importance of stating clearly at the 

 outset what stage of abstraction is being considered in his 

 analysis; conclusions which are applicable at one stage 

 may not be applicable at another. 



Since this is an important difficulty, a few remarks may be 

 made relative to the position which will be taken in the 

 following chapters. Though much is to be gained by con- 

 sidering the concepts at a high level of abstraction, much 

 is also lost. Consider, for example, the use of measurement 

 techniques in science. By this process a qualitative event, 

 e.g., a force, a duration, or a motion, is replaced by a number 

 corresponding to the reading on an instrument; the force 

 is replaced, say, by the number 5 which represents so many 

 dynes, the duration by the number 2 which represents so 

 many seconds, and the motion by the number 3 which 



