THEORIES AND MODELS 219 



liA = B 



and JB = C 



then A=:C 



Given the premises, we may readily agree that it is necessarily the 

 case that A ^ C. Assigning denotations to the abstract symbols, we 

 now let A, B, C represent the "weights of objects"; and we let the 

 symbol = stand for "equal within the sensitivity of the best avail- 

 able balance." With the assignment of these denotations it is no 

 longer necessarily the case that A = C. For example, suppose that A 

 is lighter than B by an amount barely less than the sensitivity of the 

 balance, and that B is lighter than C to just the same extent. Then, 

 although A = B and B ^ C, when we compare A and C we find A 

 clearly lighter— so that A-^C. 



Such contingency we do not ordinarily associate with logic and 

 mathematics, but of course it is not logic and mathematics that 

 then fail. We left the certitudes of logic and mathematics behind us 

 when, to acquire a scientifically meaningful statement, we invested 

 the formalism with experiential relevance. No longer vacant, the 

 formally impeccable conclusion is now no longer necessary. Ben- 

 jamin Peirce defines mathematics as "the science which draws neces- 

 sary conclusions." But the necessity of the conclusions arises pre- 

 cisely because, as Peirce's distinguished son Charles Sanders well 

 recognized, mathematics is not a science. 



The seductive misconception that mathematics is sl science may, 

 on occasion, prove very actively misleading. Thus, for example, con- 

 sider that we discover some elegant formal demonstration that B 

 "follows necessarily" from a set of premises including A. Identifying 

 A and B with certain concepts, we find that B represents a known 

 colligative relation. Betrayed by the thought that mathematics 

 teaches us something about the world, we may then fall prey to such 

 age-old fallacies as affirmation of the consequent: finding B, we con- 

 clude that A certainly exists. Incredible as it may seem, competent 

 scientists have been so deluded. A celebrated instance developed 

 from Newton's demonstration that: 



If the density of a fluid which is made up of mutually repulsive parti- 

 cles is proportional to the pressure, the forces between the particles 

 are reciprocally proportional to the distance between their centers, 

 And vice versa, mutually repulsive particles, the forces between which 



