  Toward the end of the nineteenth century, however, Peirce began to hold that Kanticity and Aristotelicity, even when conjoined, were insufficient to define adequately the notion of a continuum. He maintained that he had framed an updated conception of continua by somewhat loosening his attachment to Cantor’s ideas. He began to write in ways that, at least at first glance, seem close to falling into Cantor’s Paradox; Peirce, however, tried to avoid outright contradiction by means of embracing some sort of non-standard idea about the identity of points on a line. For example, in Lecture 3 of his Cambridge Conferences Lectures of 1898, published as Reasoning and the Logic of Things, Peirce says that if a line is cut into two portions, the point at which the cut takes place actually becomes two points. What Peirce’s new approach is, in mathematical detail, and whether or not it contains hidden but real contradictions, is a problem that has not yet been solved by researchers into Peirce’s logic and mathematics.