  First, perhaps mathematical objects exist only in a potential manner, which contrasts with the actual mode of existence of ordinary physical objects. This idea is at the heart of the ancient notion of potential infinity (Lear 1980, Linnebo &amp; Shapiro 2017). According to Aristotle, the natural numbers are potentially infinite in the sense that, however large a number we have produced (by instantiating it in the physical world), it is possible to produce an even larger number. But Aristotle denies that the natural numbers are actually infinite: this would require the physical world to be infinite, which he argues is impossible. 