Axiomatic approaches to science and mathematics depend on an underlying set of statements, principles, or propositions that apply to all situations within the domain of study. The axioms run the gamut from undisputed universal laws to widely or even universally accepted but unproved or unprovable generalizations, to propositional stipulations adopted for analytical convenience or because they raise interesting questions.

Examples abound in mathematics and formal logic, and in science, engineering and technological applications of math and logic. Although it is only occasionally referred to as such, the laws of stratigraphy (details in any geology textbook) form an axiomatic approach to sedimentology, sedimentary geology, and related palaeoenvironmental studies. The laws of original horizontality, lateral continuity, superposition, and cross-cutting relationships are assumed in this approach to apply to all sedimentary deposits, and therefore form an axiomatic system for interpretation.

A big problem with predicting responses to global climate change (or other environmental changes) is that they are nonlinear and thus disproportionate. Sometimes large changes can have relatively small responses, while in other cases small changes can have disproportionately large impacts.

Responses to environmental change are sometimes characterized by amplifiers—phenomena that reinforce or exaggerate the effects of the change. For example, if coastal land is subsiding, this amplifies the effects of sea level rise. Or, when warming results in permafrost thawing, this releases methane, a heat-trapping greenhouse gas, this leads to further warming. However, there are also filters—phenomena that resist, offset, or diminish the effects of the change. For instance, if coastal land is tectonically or isostatically uplifting, this can offset or even eliminate effects of sea level rise with respect to coastal submergence. Or, if warming results in increased cloud cover, which reflects more radiation, this counteracts the warming.

Science fiction and popular science writer Arthur C. Clarke once wrote that "any sufficiently advanced technology is indistinguishable from magic." Riffing on that theme, I once gave a talk in which I proclaimed that "any sufficiently improbable event is distinguishable from the miraculous." Some definitions of "miracle" invoke the divine or supernatural, but I have in mind the definition (in this case from the Merriam-Webster dictionary) as: "an extremely outstanding or unusual event, thing, or accomplishment." The point of the argument is that, due to the inescapable, irreducible role of geographical and historical contingency in Earth surface systems, all such systems (landscapes, ecosystems, soils, etc.) are unique in some respects (a formal argument along these lines is presented in this article: Phillips, J.D.  2007.  The perfect landscape.  Geomorphology 84: 159-169.). Thus the probability of existence of any given state of any given system at a given point in time is infinitesimally low. This exceedingly low probability makes nearly any environment in some senses extremely outstanding and unusual, and thus a miracle.