Why Arithmetic’s Incompleteness Might Suggest That Mathematics Is Discovered

Abstract

Is mathematics invented or discovered? This question has perturbed philosophers and mathematicians alike for centuries. In this manuscript, we propose a novel argument in favor that fundamental arithmetic is dis- covered rather than invented using Kurt Gödel’s Incompleteness Theo- rem because essentially incompleteness seems to be a property unique to arithmetic and counter-intuitive to invented systems. We first summarize Gödel’s argument, then argue that arithmetic’s essential incompleteness shouldn’t be a property in purely invented systems without it, giving an example on how to resolve incomplete systems in those systems and argue that arithmetic is to some extent discovered because of this difference in property. We also conjecture that any invented system that could be proven to be essentially incomplete with a similar logic to Gödel’s method can express arithmetical relationships in some form. Finally, we account for some objections to discovery.