Mathematics as Second-Order Chisel: A Philosophy of Mathematics

This paper offers a structural answer to Kant's question "How are synthetic a priori judgments possible?" within the Self-as-an-End framework: a priori = the transcendental ground is exceptionless; synthetic = the subject chisels out a new degree of freedom. Mathematics is the first concrete instance of this answer. Philosophy chisels chaos and constructs the law of identity (A=A). The law of identity is exceptionless—this is the source of the "a priori" in mathematical judgments. The law of identity automatically exposes the dimension of quantity ("more than one"); the subject exercises negation upon quantity ("two is not three") and constructs the law of non-contradiction (A cannot simultaneously be not-A)—this is the source of the "synthetic." Both chiseling and constructing are free acts of the subject (invention); the object of study (the law of identity) is exceptionlessly there (discovery). The "invention vs. discovery" debate is thereby dissolved. The paper establishes seven contributions: (1) the core thesis of mathematics as operation upon the law of identity; (2) the law of excluded middle as discrete special case of non-contradiction; (3) the ontological status of mathematical objects; (4) a structural formulation of Gödel's incompleteness theorem; (5) the boundary of exact solvability as the boundary of mathematics' exceptionlessness; (6) the in-principle impossibility of quantifying the construct's coerciveness; (7) the closure of the meta-question "Does mathematics necessarily have open problems?" Part of the Self-as-an-End Theory Series.