 In the Prize Essay, Kant undertook to compare the methods of mathematics and metaphysics (Carson 1999; Sutherland 2010). He claimed that “the business of mathematics…is that of combining and comparing given concepts of magnitudes, which are clear and certain, with a view to establishing what can be inferred from them” (2:278). He claimed further that this business is accomplished via an examination of figures or “visible signs” that provide concrete representations of universal concepts that have been synthetically defined (Dunlop 2014, 2020). For example, one defines the mathematical concept &lt;trapezium&gt; by arbitrary combination of other concepts (“four straight lines bounding a plane surface so that the opposite sides are not parallel to each  other”[1]),  accompanied by a “sensible sign” that displays the relations among the parts of all objects so defined. Definitions as well as fundamental mathematical propositions (that space can only have three dimensions, for example) must be “examined in concreto so that they come to be cognized intuitively”, but such propositions can never be proved since they are not inferred from other propositions (2:281). Theorems are established when simple cognitions are combined “by means of synthesis” (2:282), as when, for instance, it is demonstrated that the products of the segments formed by two chords intersecting inside a circle are equal. In the latter case, one proves a theorem about any and all pairs of lines that intersect inside a circle not by “drawing all the possible lines which could intersect each other within [the circle]” but rather by drawing only two lines, and identifying the relationship that holds between them (2:278). The “universal rule” that results is inferred via a synthesis among the sensible signs that are displayed, and, as a result, among the concepts that the sensible signs illustrate.