A loose analogy relates the work of Laplace and Hilbert. These thinkers had roughly similar objectives. At a time when so much of our analytic effort goes to distinguishing mathematics and logic from physical theory, such an analogy can still be instructive, even though differences will always divide endeavors such as those of Laplace and Hilbert.

The argument from analogy is examined from the point of view of Carnap's confirmation theory. It is argued that if inductive arguments are to be applicable to the real world, they must contain elementary analogical inferences. Carnap's system as originally developed (the λ-system) is not strong enough to take account of analogical arguments, but it is shown that the new system, which he has announced but not published in detail (the η-system), is capable of satisfying the conditions of inductive analogy. Finally it is shown that an elementary analysis of analogical inference yields postulates of the η-system with a minimum of arbitrary assumptions.

Recent accounts of scientific method suggest that a model, or analogy, for an axiomatized theory is another theory, or postulate set, with an identical calculus. The present paper examines five central theses underlying this position. In the light of examples from physical science it seems necessary to distinguish between models and analogies and to recognize the need for important revisions in the position under study, especially in claims involving an emphasis on logical structure and similarity in form between theory and analogy. While formal considerations are often relevant in the employment of an analogy they are neither as extensive as proponents of this viewpoint suggest, nor are they in most cases sufficient for allowing analogies to fulfill the roles imputed to them. Of major importance, and what these authors generally fail to consider, are physical similarities between analogue and theoretical object. Such similarities, which are characteristic in varying degrees of most analogies actually employed, play an important role in affording a better understanding of concepts in the theory and also in the development of the theoretical assumptions.

Analogies have been traditionally recognized as a proper part of inductive procedures, akin to generalizations. Seldom, however, have they been presented as superior to generalizations, in the attainability of a higher degree of certitude for their conclusions or in other respects. Though Bacon definitely preferred analogy to generalization, the tradition seems to me to go the other way—until the recent publication of works by Mary B. Hesse (pp. 21-28 and passim) and, perhaps, R. Harré (pp. 23-28 and passim).

In the first part of this paper it is shown that unit names, whether simple or complex, whether of fundamental, associative or derivative measurement, may always be regarded as the names of scales. In the second it is shown that dimension names, whether simple, like “[M]”, “[L]” and “[T]”, or complex dimensional formulae, may always be regarded as the names of classes of similar scales. Thus, a new foundation for the theory of dimensional analysis is provided, and in the light of this, its nature and scope are examined. Dimensional analysis is shown to depend upon certain conventions for expressing numerical laws.