This paper investigates the consequences for our understanding of physical theories as a result of the development of the renormalization group. Kadanoff's assessment of these consequences is discussed. What he called the ``extended singularity theorem'' poses serious difficulties for philosophical interpretation of theories. Several responses are discussed. The resolution demands a philosophical rethinking of the role of mathematics in physical theorizing.

This is the last in a series of three papers on the history of the Lenz–Ising model from 1920 to the early 1970s. In the first paper, I studied the invention of the model in the 1920s, while in the second paper, I documented a quite sudden change in the perception of the model in the early 1960s when it was realized that the Lenz–Ising model is actually relevant for the understanding of phase transitions. In this article, which is self-contained, (...) I study how this realization affected attempts to understand critical phenomena, which can be understood as limiting cases of (first-order) phase transitions, in the epoch from circa 1965 to 1970, where these phenomena were recognized as a research field in its own right. I focus on two questions: What kinds of insight into critical phenomena was the employment of the Lenz–Ising model thought to give? And how could a crude model, which the Lenz–Ising model was thought to be, provide this understanding? I document that the model played several roles: At first, it played a role analogous to experimental data: hypotheses about real systems, in particular relations between critical exponents and what is now called the hypothesis of scaling, which was advanced by Benjamin Widom and others, were confronted with numerical results for the model, in particular the model’s so-called critical exponents. A positive result of a confrontation was seen as positive evidence for this hypothesis. The model was also used to gain insight into specific aspects of critical phenomena, for example that diverse physical systems exhibit similar behavior close to a critical point. Later, a more systematic program of understanding critical phenomena emerged that involved an explicit formulation of what it means to understand critical phenomena, namely, the elucidation of what features of the Hamiltonian of models lead to what kinds of behavior close to critical points. Attempts to accomplish this program culminated with the so-called hypothesis of universality, put forward independently by Robert B. Griffiths and Leo P. Kadanoff in 1970. They divided critical phenomena into classes with similar critical behavior. I also study the crucial role of the Lenz–Ising model in the development and justification of these ideas. (shrink)

The first classification of general types of transition between phases of matter, introduced by Paul Ehrenfest in 1933, lies at a crossroads in the thermodynamical study of critical phenomena. It arose following the discovery in 1932 of a suprising new phase transition in liquid helium, the “lambda transition,” when W. H. Keesom and coworkers in Leiden, Holland observed a λhaped “jump” discontinuity in the curve giving the temperature dependence of the specific heat of helium at a critical value. This apparent (...) jump led Ehrenfest to introduce a classification of phase transitions on the basis of jumps in derivatives of the free energy function. This classification was immediately applied by A.J. Rutgers to the study of the transition from the normal to superconducting state in metals. Eduard Justi and Max von Laue soon questioned the possibility of its class of “second-order phase transitions” -- of which the “lambda transition was believed to be the arche type -- but C.J. Gorter and H.B.G. Casimir used an “order parameter to demonstrate their existence in superconductors. As a crossroads of study, the Ehrenfest classification was forced to undergo a slow, adaptive evolution during subsequent decades. During the 1940s the classification was increasingly used in discussions of liquid-gas, order-disorder, paramagnetic-ferromagnetic and normal-super-conducting phase transitions. Already in 1944 however, Lars Onsagers solution of the Ising model for two-dimensional magnets was seen to possess a derivative with a logarithmic divergence rather than a jump as the critical point was approached. In the 1950s, experiments further revealed the lambda transition in helium to exhibit similar behavior. Rather than being a prime example of an Ehrenfest phase transition, the lambda transition was seen to lie outside the Ehrenfest classification. The Ehrenfest scheme was then extended to include such singularities, most notably by A. Brain Pippard in 1957, with widespread acceptance. During the 1960s these logarithmic infinities were the focus of the investigation of “scaling” by Leo Kadanoff, B. Widom and others. By the 1970s, a radically simplified binary classification of phase transitions into “first-order” and “continuous” transitions was increasingly adopted. (shrink)