Abstract

Are numbers discovered or invented? We argue that numbers exist indepen- dently of humans, yet in a form that is unspeakable, as suggested by Parmenides and Wittgenstein. Numbers become speakable only when human observers provide language, symbols, and operations. Thus, numbers are neither purely discovered nor purely invented; they emerge as a fluctuation between discovery and invention. We illustrate this view with examples such as prime numbers, zero, and imaginary numbers, and provide a simple formal model of this duality.