How should we interpret past mathematicians who may use the same vocabulary as us but with different meanings, or whose philosophical outlooks differ from ours? Errors aside, it is often assumed that past mathematicians largely made true claims—but what exactly justifies that assumption?

In this talk, we will explore these questions through general philosophical considerations and four case studies: arithmetic in Book IX of Euclid’s Elements, 19th-century analysis, 18th-century geometry, and 19th-century matricial algebra. In each case, we encounter a significant challenge to supposing that the mathematicians in question made true claims. We will show how these challenges can be addressed and overcome.