Author: Dr. Jonathan Kenigson, FRSA
Mathematicians in the classical corpus tend to sympathize with Platonic epistemologies of mathematics. In Chapter V of Arithmetic, Nicomachus of Gerasa maintains that arithmetic is a pure discovery of intellect, as opposed to an intuition or a construction (619-620). The purpose of philosophical arithmetic is to determine the relations among integers for no purpose other than erudition. Arithmetic is propose to be prior to every other form of mathematics, because integers are required for all other mathematical pursuits but not vice-versa. Hence, the a-priority of arithmetic tacitly implies the same conclusion of all other pure mathematics. The knowledge of mathematical facts, however obtained, constitutes an understanding of mathematical truth. Euclid asserts the a-priority of geometry but – contrary to Nicomachus – insists upon formal, constructive proof as a prerequisite to mathematical knowledge (30-40). The knowledge of a mathematical fact without a constructive proof is insufficient to stand in possession of a knowledge of pure mathematics. To use Platonic language, the fact without the cause lies in the domain of mixed mathematics and is sufficient only for possession of contingent (e.g. methodological) truth. In Book I of the Revolutions of the Heavenly Spheres, Copernicus defends a Platonic ontology of number as being prior to and descriptive of physical reality (510-512). Pascal’s Geometrical Demonstrations is similarly disposed, and for Pascal, the mathematician’s defining quality is the ability to reason clearly and comprehensively from first principles (axioms) that are deemed self-evident (445). Section I of Pensees defends the more general claim that mathematical practice is a process of continued discovery (or ‘revelation’) of universal facts (Pascal 444-445). Throughout his corpus, Descartes defends the a-priority of mathematical entities and furnishes a methodology for the unification of algebra and geometry to discover the properties of these entities. For Descartes, clear reasoning from self-evident first principles is the content of mathematical discovery, as established most succinctly in Meditation V (319-322). Assumptions are supposed to reflect the perceived structure of the world; deductions are intended to be judiciously systematic. The Cartesian repudiation of Aristotelian syllogism is not an abolition of logical structure but rather of what is perceived as Aristotle’s abstruse universe of axioms and byzantine hierarchy of causes. An eidetic mathematics emerges unscathed from the rebuke.
20th-century mathematical physics realizes the complex interdigitation of language and reality and – as such – represent a repudiation of Platonic epistemology. In his Principles of Relativity, Einstein argues that mathematical statements are neither true nor false, but rather reflect valid or invalid deductions from systems of axioms (195-196). These axioms ultimately derive from empirical experience, and can be discarded or modified if they fail to meet the mathematical needs of physical theories. Borrowing Kant’s language, Einstein’s mathematical statements could be called a-priori statements about a-posteriori axioms (195). Relativity represents the fruit of this program. By discarding Kantian claims of the a-priority of Euclidean geometry in favor of modern differential geometry, Einstein is able to construct a theory that supersedes Newtonian mechanics both in rigor and predictive power. As for Copernicus and Bacon, mathematics finds its ultimate utility as a bondservant of the physical sciences.
Like Einstein, Wittgenstein recognizes that mathematical statements cannot be deemed ‘true’ or ‘false’ in the absence of a language-dependent context (437). Mathematical statements are derivative of language, but are nonetheless necessarily descriptive of the physical world. The statements of mathematics are thus non-propositional but necessary; they could not, by merit of the structure of the world, be different than they are (Wittgenstein 437-438). Because of the Platonic thesis that mathematical reality is mind- and language-independent, Wittgenstein can be taken to reject Platonism, but not in-toto. The structure of language bears a necessary isometry to the structure of the world (Wittgenstein 440). To use Plotinian terms, the logos is spoken and it is infinitely reflective of the universe it generates. It is language that bequeaths the gift of mathematical knowledge, and it is the community of speakers that gives language its relation to the world. For Wittgenstein, just as there is no private language, there can also be no private mathematics. The logos cannot be spoken alone.
Although it is only infrequently appreciated, the mathematician Poincare adopts a profoundly similar view. Mathematical entities are, sui-generis, creatures of language; but their eternal predictions are not subject to revision at the mercurial whims of linguists or philosophers (1-5). If mathematical entities reflect reality, it is not because they impose their agency upon physicality, but because physicality imposes its demands upon language. As for Wittgenstein, mathematics is a semiotics of the real. Mathematical facts are necessary truths of language – not of intuition, or of reduction to logical systems (Poincare 14-15). The business of reducing mathematics to an essential methodology is as hapless as imposing a universal structure upon spoken, human languages. Mathematical truth is ultimately truth about reality, but not about the hypokymenon. The universe can disclose itself only so precisely through language, and many things of value and substance may eternally defy precise statement or description (Poincare 21-26). Einstein’s and Wittgenstein’s views that mathematics is a function of language have strong support in the Aristotelian corpus. Aristotle argues that the forms of mathematical objects inhere in the objects of description (Metaphysics 270). Mathematical objects are objects qua objects; they represent the most general possible exploration of the universe of phenomena. Syllogism, which is the science of formal language, is the universe in which mathematical objects can be said to inhere and interact with physical reality (Aristotle 547-548). Indeed, the very distinction between physical and mathematical reality is – for Aristotle – a categorical error, and a dangerous one. Platonic epistemology separates abstract reality from physicality only to arrive at the epistemological paradox that knowledge of mathematical reality is grossly mysterious. The mathematician’s art must be akin to the conjurer; his knowledge is specific and sure, but is obtained from beyond the realm where the senses – and thus description, can be reliably held to operate.
Sources.
All Citations are taken from the 1952 edition of the Great Books collection. The Syntopicon is freely available due to the generosity of the Library of India.
Hutchins, Robert Maynard. Great books of the western world. William Benton, Publisher: Encyclopaedia Britannica, 1952.
Daily magazine for entrepreneurs and business owners
