When does a heap become a heap?

Abstract

When you add one grain of sand to another, you clearly cannot call these two grains a heap. Now consider what happens if this process is repeated multiple times: can adding one additional grain of sand ever turn something that is not a heap into a heap? This classical paradox of the heap illustrates a problem arising from vague predicates and is closely related to the continuum fallacy (also known as the fallacy of the beard), which is the argument that two conditions cannot be considered distinct because between them exists a continuum of states. In the field of multiple sclerosis, we are faced with this situation when we are trying to analyse the transition from relapsing-remitting to secondary progressive multiple sclerosis (SPMS): while we all agree that this happens in a continuum, most of us would also agree that SPMS exists as a category. Hence, when we want to examine the conversion to SPMS in clinical studies, we are forced to approximate the impossible, that is, defining a distinct number of grains of sand that turns a non-heap into a heap, simply because we need a definite date of SPMS-onset for our statistical models.