The Vanishing Point: A Left-Hand Path Reading of Philosophy by Denys Spirin - HTML preview

Chapter 11

Any rung can be deflated by a standard shift: we show that the substrate is a product of the subject's operations, and the ontologization loses its grip. Mathematics resists longer and harder than anything else. Philosophers who deflate other rungs routinely keep some version of mathematical Platonism, and many who cannot justify it will still admit, when pressed, that they cannot quite let it go.

Mathematics was ontologized early, in the Pythagorean tradition. All is number.

Numbers were the substance of things, and the cosmos was constructed out of arithmetic and geometric relations the way the Milesian cosmos had been made of water or air. The harmonic ratios discovered in music, the regularities of as-tronomical cycles, the proportions of geometric figures were taken as disclosures of the numerical fabric from which everything was woven. The school extended the idea into mysticism: specific numbers acquired meanings and cosmic functions, and a full numerology developed that persisted through the Neopythagoreans and into late antiquity. Some of this survived into the philosophical mainstream and some into the esoteric tradition, where it continues to live to this day.

Plato inherited the Pythagorean idea and integrated it into his metaphysics.

Mathematical objects occupy an intermediate position in the divided line of the Republic: more real than sensible things, less real than the forms, grasped by a faculty between opinion and pure intellect. The Timaeus constructs the physical cosmos from geometric solids, with the five regular polyhedra corresponding to the elements and to the cosmos itself. A late-antique tradition holds that the Academy's entrance bore the inscription "Let no one ignorant of geometry enter." Mathematics in Plato is carved into the second-rung structure as the rigorous end of the domain of forms.

The Platonic-Pythagorean synthesis supplied the default framework for mathematical ontology. Medieval thinkers inherited it through Augustine, Boethius, 45

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and the quadrivium; the scientific revolution reinforced it. Galileo's remark that the book of nature is written in the language of mathematics is a clear fourth-rung ontologization, with the mathematician deciphering a text already written. Newton's Principia proceeded on this assumption. Descartes's analytic geometry, by translating geometric objects into algebraic equations and vice versa, made the fourth rung's structure visible in a new way and further strengthened the conviction that mathematics had access to the real.

Kant attempted a deflation. Mathematical judgments are synthetic a priori, grounded in the pure forms of sensibility — structures of the cognizing subject.

The substrate is relocated into the subject, and the necessity ascribed to it is preserved intact. What changes here is the material (mathematics rather than the categories) and the specific vulnerability: the discovery that non-Euclidean geometries describe physical space better than Euclid's undermined the Kantian framework and forced the question of mathematical ontology open again.

The foundational crises of the late nineteenth and early twentieth centuries produced three explicit answers.

Logicism, developed by Frege and carried forward by Russell and Whitehead in the Principia Mathematica, held that mathematics is reducible to logic. The numbers and structures of mathematics are definable in logical terms, and mathematical truths are logical truths in disguise. Frege's drittes Reich, men-tioned earlier as a second-rung ontologization of thoughts, is the same domain that houses mathematical objects. For Frege the second and fourth rungs are continuous — logical and mathematical truths inhabit a single objective realm, and the reduction of mathematics to logic is at the same time the unification of the two rungs into one.

Formalism, associated with Hilbert, took the opposite direction. Mathematics is the manipulation of symbols according to rules. Mathematical objects have no existence beyond the formal systems in which they appear; the question whether numbers really exist is a confusion about what mathematics is. Mathematical truth is consistency of a formal system, and the task of foundations is to prove the consistency of arithmetic and analysis within a finitary framework.

The ontologization of the fourth rung is replaced with a theory of practice.

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Gödel's incompleteness theorems showed that the finitary consistency program cannot be completed in the form Hilbert hoped, but the formalist attitude survived and is the default working position of most practicing mathematicians, who proceed as if formalism were true and talk as if Platonism were true.

Intuitionism, developed by Brouwer and codified by Heyting, returned mathematics to the subject. Mathematical objects are mental constructs, and a proposition is true only when it has been constructed. There are no mind-independent mathematical facts; there are only the acts the mathematician performs.

Classical logic, including the law of excluded middle for infinite domains, is rejected as it presupposes that every mathematical proposition has a determinate truth-value independent of construction. The ontology of mathematics becomes the ontology of constructive acts. This recognizes that mathematics is produced by the subject and also, characteristically, stops before completing the analysis. The mental constructions Brouwer invokes are treated as a bedrock, and the question of how the subject produces them and how the products receive their stability is left unsolved.

Later twentieth-century philosophy of mathematics multiplied the positions.

Mathematical structuralism, developed by Resnik, Shapiro, and others, holds that mathematical objects are positions in abstract structures, with no identity apart from their relational role within the structure. This is a sophisticated formulation but remains ontologizing — the structures themselves are taken to exist and to be the subject matter of mathematics. Nominalist programs, no-tably Hartry Field's, attempted to dispense with mathematical objects entirely by showing that discourse can be reformulated without commitment to numbers or sets; science uses mathematics as a useful fiction whose indispensability Quine had famously taken as an argument for mathematical realism. Fiction-alism, constructivism in its various forms, and category-theoretic foundations have each produced further variations. The space of positions is large and the disagreements are technical and often intricate.

Across the variations, the system of mathematical codes is either ontologized directly (Platonism in its various forms) or relocated (logicism, formalism, structuralism, intuitionism). A relocation stops at a new unexamined layer —

the pattern traced on earlier rungs repeats here with particular tenacity.

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The reason mathematics resists deflation more than the other rungs is worth stating directly. Mathematical distinctions are maximally stable. Two plus two gives four for any subject at any time of the day. The recursive distinctions converge on results that depend so little on the mathematician that the subject appears to drop out entirely. This convergence is the best possible case for an ontologizing reading. If any rung has a claim to mind-independent existence, it is this one.

This invariance follows from the depth of recursion, not from residence in an independent domain. Mathematical distinctions are what remains when every contingent feature of the distinguishing operation has been abstracted away.

Any distinction at all produces two: the distinguished and what it has been distinguished from, A and not-A. Apply the operation to the result and a third appears, then a fourth, and so on. Arithmetic is the internal structure of the distinction, made visible when it is turned upon itself. Spencer-Brown worked this out explicitly in Laws of Form, deriving the propositional calculus and its equivalent two-valued Boolean algebra from the single primitive act of drawing a mark. Dedekind and Peano had shown the same, defining the natural numbers through iteration of the successor function whose content is nothing apart from applying the operation again.

What holds for arithmetic works for the rest of mathematics by extension.

Geometry emerges when distinctions are drawn in extension and their relations are preserved under operations that respect those relations. Algebra arises when the patterns of iterated operations are distinguished and compared. Each higher mathematical domain is a further recursion.

This answers the puzzle that the fourth rung resists deflation more stubbornly than the others. The other rungs contain products of distinction applied to specific materials — qualities to perception, concepts to experience, language to practice, laws to measurement. Their content is contingent on the material. The products of the fourth rung have no such contingent ingredient. Every subject already has them in virtue of being the one who distinguishes. When they are reflected upon, it finds structures that are stable across other observers, and reads this stability as mind-independence.

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This answer is not likely to convince a working mathematician, and it does not need to. A mathematician can proceed on Platonist assumptions and do his work; the ontological question is orthogonal to the technical one. What the answer does is block the move by which Platonism is recruited to support ontological claims on other rungs — the argument that if numbers really exist, then Platonism is true and this proves the existence of God, for example. The fourth rung has been used as a battering ram from the Pythagoreans through the scientific realists.