Dr. Jonathan Kenigson, FRSA

Dialogue regarding mathematics falsely assumes that mathematics is a subject. A subject as traditionally defined requires a concrete domain of discourse and a methodology for establishing or negating truth-claims. For so-called mathematics, this domain is taken to be "numbers, functions, and the like," and the operant methodology is taken to be deductive reasoning. Even Nominalists (Hartry Field included), for instance, tacitly define mathematics to possess a certain domain of discourse consisting of an amorphous range of entities "like" numbers and functions. One must inquire which form of similitude is implied and whether the mark that denotes "2" is like "two" in the same sense that two birds are like two stones, or two grams are like two cars. A function "is" a relation consisting of ordered pairs of numbers, so how is the relation unlike the number? The constellation of questions surrounding similitude in a mathematical sense are only infrequently reducible to mathematical questions. Maths and philosophy parted ways in bad company and left a series of paradoxes in the middle.

It is difficult to say how "three" differs from a closed polygon with interior angles summing to pi, or a Black Hole differs from a "spacetime singularity with escape velocity equal to c." The geometric presupposes the arithmetic, and the astronomical supposes both of these in addition to motion. Music presupposes all these elements in harmony and proportion. To say that the fourth note is like or unlike the half, or that the harmony of notes is most pleasant in some ratio, presupposes all of number, figure, motion, and harmony at once. It is for this reason that music is maths "per-se." The Western world gradually forgot this truth during the Middle Ages and recovered it in the Renaissance, only to lose it again in Descartes' Geometry and Hegel's transcendental rationalism. Spinoza's rationalism recovered - for a historical moment - the interdigitation of Number and Nature in the form of Monadology. Sadly, the Monad had to be sacrificed to the formalism of the "Principia" to provide a Principle of Sufficient Cause formulated in Analytic-Algebraic language. Such language promised to render understandable change as eternal and immutable via Dynamical Systems and Differential Equations.

Deterministic certitude - present in Newton's Mechanics and Fourier's mathematics - is an excellent foundation for mathematical physics but a poor foundation for physical mathematics. Triangles do not exist in nature, but things "like" triangles in an intuitive sense exhibit "triangle-like" properties. Their interior angles sum to close to pi, their lengths roughly obey the Laws of Sines and Cosines, and their trigonometry is roughly approximable by periodic functions. The Scholastics could unite Physics and Metaphysics under the aegis of the Syllogism, but the philosophers of the secular Renaissance could not capture a reason why there is a strong physical correspondence between truths reasoned from axioms and truths observed in nature. Islamic philosophy never required such a distinction. All later Structuralism, Nominalism, Platonism, and Functionalism have sought to restore the certitude that Aristotle wagered he had gained in rebelling against the Forms or, more pathologically, to establish that no such certitude is possible.