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The science of value, or value science, is a creation of philosopher Robert S. Hartman which attempts to formally elucidate value theory using methods which are, or are claimed to be, mathematical. The fundamental principle, which Hartman calls an "axiom", is that a thing is good insofar as it exemplies its concept. This means, according to Hartman, that the good thing has a name, that the name has a meaning defined by a set of properties, and that the thing possess all of the properties in the set. A thing is bad if it does not fulfill its definition. A car, by definition, has brakes. A car which accelerates when the brakes are applied is a bad car, since a car by definition must have brakes. A horse, if we called it a car, would be an even worse car, with fewer of the properties of a car. He introduces three basic dimensions of value, systemic, extrinsic and intrinsic, each with their own cardinality for sets of properties; finite, <math>\aleph_0<math> and <math>\aleph_1<math>. The terms "good" and "bad" apply only to finite sets of properties, since this is the only case where there is a ratio between the total number of desired properties and the number of such properties possessed by some object being valued. In the case where the number of properties is countably infinite, the extrinsic dimension of value, the exposition as well as the mere definition of a specific concept is taken into consideration.
Hartman proposes to quantify this notion by the principle that each property of the thing is worth as much as each other property, depending on the level of abstraction. (The Structure of Value, page 204.) Hence, if a thing has n properties, each of them -- if on the same level of abstraction -- is proportionally worth 1-n. In other words, a car having brakes or having a gas cap are weighted equally so far as their value goes, so long as both are a part of one's definition of one's personal concept of a "car." Since a gas cap is not normally a part of a car's definition, it would be given no weight. Headlights could be weighed twice, once or not at all depending on how headlights appear in the description of a car. Given a finite set of n properties, a thing is good if it is perceived to have all of the properties, fair if it has more than n/2 of them, average if n/2 of them, and bad if it has fewer than n/2.
Hartman goes on to consider infinite sets of properties. Hartman claims that according to a theorem of transfinite mathematics, any collection of material objects is at most denumerably infinite (The Structure of Value, page 117.) This is not, in fact, a theorem of mathematics, though it would follow from certain assumptions on the nature of the physical universe which cosmologists typically make. Starting from the claim that a person can eventually think of a countable infinity of things, Hartman concludes the intension of man is a denumerably infinite set of predicates; which means that man, according to this first definition, is appropriately to be measured by a denumerable infinity. However he quickly passes to the conclusion that we also have a countable infinity of levels of thought, and that therefore we can think of a countable infinity of things using a countable infinity of thought levels, giving us the cardinality of the continuum of thoughts. Hartman believes the generalized continuum hypothesis is true, and therefore claims the intension of man consists of <math>\aleph_1<math> elements. This is the cardinality, in Hartman's system, of intrinsic value.
Further combinations are possible, leading to larger uncountable infinities; and Hartman also introduces the reciprocals of aleph numbers, which play no role in ordinary mathematics, but which Hartman employs as a sort of infinitesimal proportion, and which he contends goes to zero in the limit as the uncountable cardinals become larger. In Hartman's calculus, a Dear John letter ("we will always be friends") has axiological value <math>\frac{1}{\aleph_2}<math>, whereas Christmas shopping or taking a metaphor literally would do better, with a value of <math>\frac{1}{\aleph_1}<math>.
Evaluation of Hartman's work
Hartman's work led to the Hartman Value Profile, which is however not a description of what is valuable, but a psychological test to determine what people regard as valuable, and which therefore belongs to psychology as a disciple. A Hartman festshrift (Values and Valuation) appeared a few years after his death, but most of the articles in it are not by Hartman supporters, and some of them are by people who are known to be critics of his work. Hartman is out of the mainstream of value philosophy, and is not regarded as a significant presence. Almost all philosophers would dispute the idea that the number of properties of a thing can in any meaningful way be enumerated, and a standard argument against that is that new properties can be defined in terms of old ones. Philosophers speak of the problem of organic value as a result of the observation that the value of a whole does not seem to be a mere sum of the value of its parts, however we define parts or value, and that adding more features, even if each seems to be a good one, can lead to the overall value going down. In this way we get overengineered software or the kind of DVD remote control which has too many buttons on it.
From a mathematician's point of view much of Hartman's work is highly dubious, and certainly not an instance of correct mathematical methodology or axiomatic reasoning. Hartman uses infinite cardinalities, but fails to mention his assumptions, to define his reciprocals of cardinal numbers [however see superreal number to understand why such cardinals are mathematically legitimate], or to clearly explain how he calculates the value of such items as Christmas shopping in terms of them. Even when the arithmetic is correct the motivation seems to be lacking; an example is adding together good, fair, average and poor. The conclusion that this comes to two and a half times good is merely an artifact of the definition of these categories, and moreover requires us to add the satisfied properties of a fair object to the satisfied properties of the complementary poor object, a fact which Hartman apparently fails to grasp, and which seems to make no sense. Unless some reason is supplied as to why any of this makes sense or has some purpose, we lack an important element required for a mathematical model.
Hartman supporters counter that it is not necessary for properties to be actually enumerated, only that they exist and can correspond bijectively. The attributes in the meaning of a concept only "consist" as stipulations; they don't exist; actual existence of an exemplar of such a concept belonging to ontology. They can resemble, but are not identical to, the properties perceived by the five senses.
References
Hartman, Robert S., The Structure of Value: Foundations of Scientific Axiology, Southern Illinois University Press, 1967
Hartman, Robert S., "Axiometric Structure of Intrinsic Value", Journal of Value Inquiry (Summer, 1974; v.8, no. 2, pp. 88-101
Katz, Marvin C., Sciences of Man and Social Ethics, Boston, 1969, esp. pp. 9-45, 101-123.
Davis, John William, ed, Value and Valuation: Axiological Studies in Honor of Robert S. Hartman, The University of Tennessee Press, 1972
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