Alien Mathematics: Notation and the Thinkable. How Numeral Systems Shape What Mathematics Can Imagine

A numeral system is often treated as a transparent window: a neutral way of writing numbers that are themselves unchanged by how they are written. This paper argues that the window is not transparent. Notation does not determine mathematical truth, but it strongly shapes what is easy to compute, easy to discover, easy to manipulate, and easy to regard as mathematically real. Historical vocabulary preserves traces of this friction: negatives were called absurd, irrationals alogical or unspeakable, and the sqare root of $-1$ imaginary. These names are read here not as accidental curiosities but as markers of moments when mathematical structure outran available representational tools. Two focused thought experiments sharpen the claim. The Babylonian sexagesimal system shows how a sophisticated positional notation could remain structurally limited by the absence of a true zero and by the silent effects of base. Roman numerals, considered against the history of proto-calculus and early modern symbolic advances, show that notation may leave logical truth intact while placing discovery, manipulation, and routinisation effectively out of reach. The resulting lesson is methodological: notation is best understood as a cognitive technology, not as a transparent medium. If so, the history of mathematics becomes in part a history of notational reach, and the final question is unavoidable: which structures might current notation still be hiding from us.