linguistically differentiated. The systematic constraints involve adjusting mathematical representations, physical meanings, and general laws to improve the overall coherence and adequacy of physics. When quantitative concepts are widely accepted and used in practice, then they are generally treated as facts about the world, rather than as parts of an inferential system.
To implement one aspect of this network model we will consider the co-evolution of physical and mathematical concepts. The concept of a quantity is intrinsically related to mathematical structures. As Piaget has convincingly demonstrated, a growing child does not really grasp the concept of a quantity until she realizes that quantity is invariant under simple transformations. Pouring milk from a tall thin glass into a short squat glass does not change the quantity of milk. In the thirteenth century the notion was extended to the quantity of a quality, first the non-physical quality of sanctifying grace, later to quantity of motion. Following Nicole Oresme and Descartes this could be represented by areas which are invariant under change of shape, and later by numbers.
The further extension to heat illustrates one stage in the mutual adaption of physical concepts and mathematical forms. After the development of thermometers various formulas for mixing substances with different temperatures were offered. The one that won general acceptance was Richman's, = (m1t1 + m2t2)/ (m1 + m2), where m1, t1, m2, t2 correspond to the masses and initial temperatures of the two substances mixed together and to the resultant temperature of the mixture. (See McKie and Heathcote). This fit selected experiments involving similar substances, no changes of state, and sloppy measurements. However, it could not accommodate two types of data. The first was that the same amount of heat (judged by length of heating under standard conditions) led to different temperature rises for different substances of the same mass. The second was that under some conditions the application of heat led to a change of state, rather than a rise in temperature. Black and Wilke, more or less independently, overcame these difficulties by introducing the concepts of specific heats of substances, overt and latent heat, and specific heats of vaporization and freezing. The new and now familiar mixing formulas, where no change of state is involved, balanced the heat gained by a fluid with mass, m1, and specific heat, c1, at an initial temperature, T1, in a beaker of mass, m2, and specific heat, c2, also at initial temperature, T1, by the heat loss by a substance of mass m3, specific heat, c3, an initial temperature, T2, where T2 > T1. When the substance is immersed in the fluid and the beaker is thermally insulated then the final temperature, Tf, is given by the formula:
[m1c1 + m2c2] (Tf – T1) = m3c3 (T2 – Tf) (7)
Such formulas have properties that were gradually recognized as characterizing quantities. They must be specified in a way that admits of serial ordering and some physical process of concatenation corresponding to addition. A functionally adequate concept of the quantity of heat required embedding this concept in a network of related concepts and practices. As physics advanced, the network of concepts was refined and the mathematical formulations improved. Specific heats were found to be a function of temperature. The concept of temperature, as noted earlier, was finally defined in a way that did not depend on the properties of any particular substance. The key idea behind (7) is the conservation of a quantity. Black simply interpreted it as quantity of heat transferred. Others, notably Lavoiser and Laplace, interpreted it in terms of conservation