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Mathematics, in its operations with variable quantities, enters the field of dialectics; and it is significant that it was a dialectical philosopher, Descartes, who introduced this advance. The relation between the mathematics of variable and the mathematics of constant quantities is in general the same as the relation of dialectical to metaphysical thought. Engels, Anti-Duhring, Part I Philosophy, Chapter 12, Dialectics, Quantity and Quality . . . The primary subject matter of mathematics is still the variation of quantity in time and space, but that also this primacy is partly of the nature of a "first approximation", is reflected in the increasing importance of structures (not only of quantities) in the mathematics of the last 100 years. For example, a first approximation to a theory of a material situation involving three apples might be simply the number (constant quantity) 3. The idea of an abstract set of three elements is a somewhat more accurate theory. If one of the apples happens to be distinguished, for example, by being green, we may consider the simple structure of an abstract set with a distinguished element, a theoretical refinement which the quantity three does not really admit; the unique non-trivial automorphism of this structure is a theoretical operation which again the quantity itself does not admit. This simple example indicates that at least in some cases the idea of a structure (representing a quality) is a refinement of the idea of a quantity. The abstraction from structure/quality to quantity (obvious in the above simple example) is present and is significant also among variable structures and variable quantities. . . . Platonic idealism, while mistaken on the relation between consciousness and matter, contains an aspect of truth when applied within the realm of ideas. For example, the doctrine of algebraic theories begins with the affirmation that there exists as a definite mathematical object the "perfect idea of a group" of which the idea of any particular group is "merely an imperfect representation". Indeed, it is a simple further step to form as a mathematical object the "idea of an algebraic theory" of which any particular algebraic theory is a representation, as was carried out as a part of Benabou’s 1966 dissertation. Of course, even here Platonic idealism is wrong on the order of development, since the ideas of several particular groups were concentrated from practice well before the idea of a group in general was concentrated from the practice of mathematicians around 1800., F. William Lawvere (1972) Perugia Notes: Theory of Categories over a Base Topos, Universita di Perugia, Perugia.

An adequate subcategory isa subcategory of a category such that maps whose domains are in the subcategory are adequate to determine every object of the category. Dually, a coadequate subcategory is a subcategory such that maps whose codomains are in the subcategory are coadequate to determine every object in the category (Lawvere (1994) Tools for the advancement of objective logic: Closed categories and toposes, in The Logical Foundations of Cognition, J. Macnamara andG. E. Reyes (Eds). Oxford University Press, p. 49).

The research reported in this thesis was an exploratory nature undertaken for the purpose of developing a measure of the adequacy of health services in three rural counties in Utah. The three counties included in the study area were Beaver, Piute, and Wayne. Adequacy of health services in the three-county area was measured on the basis of (1) the general population's, or consumer's, perception of the adequacy of the existing services, (2) the professional's or provider's perception of the adequacy of the existing services, and (3) adequacy of the existing services determined by a comparison with an established standard. The findings showed that major differences in the general population's perception of adequacy are present when controlling for county with the vast majority of the respondents in Piute and Wayne counties perceiving the existing services as poor, but with a large majority of those in Beaver County considering their services as good or excellent. With regard to the professional's perception of adequacy of existing services, it was found that the majority of the medical doctors perceived the services as adequate, but the doctors of dental surgery generally perceived the adequacy of service as somewhat less than desirable. With regard to the last measure, that of adequacy of existing services as determined by a comparison to an established standard, the services in the study area were found to be inadequate in all respects with the single exception of number of hospital beds per 1,000 population in Beaver County. It was concluded that there is a large degree of consensus between the perceived inadequacy of the health services of the general population in Piute and Wayne counties and the inadequacy demonstrated by the comparison of existing services to an established standard. In Beaver County the reverse is true with the perception of the general population and that of the physician population being very similar, both considering the services adequate but both being in disagreement with adequacy as measured by an established standard. The findings suggest that more detailed research in the area is needed if a standard of adequacy acceptable to everyone is to be developed.