Punctuated Equilibrium
Along with paleontologist Niles Eldredge, Stephen developed the concept of punctuated equilibrium in order to explain a feature of the fossil record; that there were apparent bursts of speciation with long periods of little or no change in between. According to the two researchers, environmental stresses would frequently be low enough that there would be no reason for evolutionary changes, but when significant environmental shifts did occur, evolutionary change happened, and sometimes happened relatively rapidly. There were public disagreements between Richard Dawkins and Stephen about punctuated equilibrium, though Richard's arguments were more about Stephen's effort to make a distinction with Charles Darwin's expectation of a steady gradual evolutionary rate than with the basic ideas behind punctuated equilibrium.
Another important concept in evolution that Stephen advocated and popularized was that the evolutionary process does not have a predetermined outcome. If a portion of the evolutionary record were to be replayed, some large-brained, intelligent species would probably still exist, but the species might not be human and might not even evolve from the line of primates. One of his most famous quotes is “Humans are not the end result of predictable evolutionary progress, but rather a fortuitous cosmic afterthought, a tiny little twig on the enormously arborescent bush of life, which if replanted from seed, would almost surely not grow this twig again.”
Non-Overlapping Magisteria
In the book Leonardo's Mountain of Clams and the Diet of Worms, Stephen explains that the standard attitude of all major Western religions and of Western science today is that science and religion are different and compatible. "The lack of conflict between science and religion arises from a lack of overlap between their respective domains of professional expertise - science in the empirical constitution of the universe, and religion in the search for proper ethical values and the spiritual manning of our lives. The attainment of wisdom in a full life requires extensive attention to both domains - for a great book tells us both that the truth can make us free, and that we will live in optimal harmony with our fellows when we learn to do justly, love mercifully, and walk humbly."
Although Stephen was a Jew, he draws from the teachings of the Roman Catholic church and the papal statements of Pope Pius XII and Pope John Paul II, which refer to the Magisterium, or teaching authority, of the church in regards to a potential conflict between science and religion. "No such conflict should exist because each subject has a legitimate magisterium, or domain of teaching authority - and these magisteria do not overlap. The net of science covers the empirical realm: what is the universe made of (fact) and why does it work this way (theory). The net of religion extends over questions of moral meaning and value. These two magisteria do not overlap, nor do they encompass all inquiry. ... To cite the usual cliches, we get the age of rocks, and religion retains the rock of ages; we study how the heavens go, and they determine how to go to heaven." He also writes that "non-overlapping magisteria represents a principled position on moral and intellectual grounds, not a merely diplomatic solution. Non-overlapping magisteria also cuts both ways. If religion can no longer dictate the nature of factual conclusions residing properly within the magisterium of science, then scientists cannot claim higher insight into moral truth from any superior knowledge of the world's empirical constitution. The mutual humility leads to important practical consequences in a world of such diverse passions."