Author: Dr. Jonathan Kenigson, FRSA
Author Affiliation: Athanasian Hall, Cambridge LTD (UK, USA)
The national news has recently been awash in debate about Hillsdale College’s philosophy of classical mathematics instruction and its potential applications in private and charter-schools. It is the contention of the current author that concrete exploration of the classical methodology is indispensable to the contours of the debates. In particular, this discourse proffers a concrete example of classical mathematical methodology in terms of a brief introduction to classical education applied to the study of mathematics. Several key themes should be clearly and unequivocally noted. First, one need not have an inherently Christian (and certainly not a Protestant Christian) basis to a Hillsdale-style classical liberal arts curriculum. However, it is important not to divorce mathematics from the cultural milieus in which it is produced and negotiated. Because Christianity has exerted a substantial influence in Western thought, one must fairly include Christian thinkers in the philosophical discussion of mathematical topics.
Additionally, it is important for proposed practitioners of the Hillsdale methodology to understand that deep discussion and analysis of primary sources is necessary for the full force of the classical model to be realized. Mathematics textbooks should be considered anathema to the classically-trained mathematician. Principles must be adduced from careful attention to the texts and traditions that inform the history and philosophy of mathematics. These do not exist in the form of endless exercises and mindless, rote repetition. They similarly do not inhabit the dreary world of the impersonal learning-management systems that students so completely and rightfully loathe. Perhaps the greatest disservice to any claim of a classical mathematics “curriculum” is the supposition that such thing can exist at all. Classical methodologies of mathematics are inherently philosophical in nature and focus foremost on the existence and utility of the quantities typically studied (purely formally) in mathematics curricula. In a word, one can have no classical mathematics – Hillsdale or otherwise – that is not informed primarily by a philosophical evaluation of mathematical practice.
The current discourse is an example of Quadrivium-style mathematical work that students in a classical school or college should and must undertake. As such, it represents a culmination of several distinct trajectories of philosophical and mathematical research, each of which merits independent deconstruction. From the epistemological end, the object of examination is mathesis – or, more commonly, the manner in which new mathematics is synthesized from that which already exists in the research literature. The goal of the inquiry will be to demonstrate that mathesis represents a self-conscious process of mathematical generalization, in which many mathematical practitioners purposively seek to make their results as widely applicable (general) as possible. One result of this generalizing tendency is a counter-claim to the Platonist thesis – most prominently the so-called Quine-Putnam Thesis (QPT) – that mathematics is a-priori predictive of physical reality and that abstract objects must exist to explain this predictive power. In anticipated future submissions, we shall follow Quine in determining mathematical theories to be reflective of symmetries inherent in empirical reality, and also verify Wittgenstein’s thesis that mathematics represents a sort of ‘language’ accounting for the symmetric properties of physical reality.
A classical tacit epistemological assumption is that mathematics is the artifact of its discourse – proofs, theorems, lemmata, and published results – rather than the intuitions and habits of its practitioners. The study of how mathematics is done prior to the production of artifacts is considered unimportant to the more intractable questions in the epistemology of mathematics. Philosophy of mathematics is thus an archaeology rather than an anthropology. Contemporary epistemology of mathematics consequently eschews the history of mathematical thought from ancient to early modern times, which are crucial epochs in the co-evolution of mathematics and philosophy as a disciplines. The characters of the discourse, history of the problems, intuitions of the disputants, and requirements from allied disciplines make no entrance into the analysis. Another aspect of the debate is that mathesis – in addition to being a practical mode of mathematical production – is the presence of alternating Platonistic and Aristotelian tendencies in the philosophy of number, quantity, and relation that occur in the classical philosophical corpus. We shall thus be arguing that mathesis is not only a mathematical phenomenon but rather a dialectic between Platonic and Aristotelian zeitgeists in the philosophy of mathematics, mechanics, and the empirical sciences. Such interdisciplinary work ought to be welcomed by traditional philosophy of mathematics.
The dearth of work combining philosophical and mathematical insights leads working mathematicians to equate mathesis with its artifacts (proofs) so that the epistemology of mathematics is to a great degree subordinated to proof per-se. A foray into the abstract philosophy of mathematical discovery should represent a demonstration of how mathesis operates across broad domains of mathematical research and how it can avoid quantification over the abstract objects of QPT while remaining physically explanatory and predictive. The nature of mathematical truth has been debated by philosophers, historians, and mathematicians alike. This interest has been complemented by the study of the origins and foundations of mathematics. Predictably, among historians, there is an emphasis upon the search for the sources of mathematical practice. For instance, in Book 2 of his Histories, Herodotus asserts that the origins of geometry lie in Egyptian agricultural applications; the Decline and Fall of the Roman Empire also explores the foundations of geometry.
Platonism is the historically predominant perspective in the philosophy of mathematics. In the Gorgias, Plato divides the arts into those that can be done silently and those that demand verbal arguments (252-295). More importantly for our discourse, the same dialogue establishes a distinction between pure (“philosophical”) and applied mathematics – with the former closer to pure dialectic than the latter (Plato 254). Because Gorgias is primarily focused upon the nature of rhetoric, further direct discussion of mathematical topics is limited (Plato 451). On the other hand, the Timaeus is intimately related with the mathematical order of the cosmos and provides a wide-ranging account of the mathematical nature of reality. The demiurge instantiates ‘kosmos’ (ordered reality) from the world of forms, and it is the very purpose of the universe to disclose its mathematical nature (Plato 254-255). The arithmetician, like the geometer, is engaged in a disclosive and exploratory – as opposed to a constructive – process. The Philebus is devoted to the study of pleasure and knowledge; this late dialogue further delimits the superiority of pure mathematical knowledge to ‘mixed’ (e.g. applied) knowledge. Thus, among the later platonic dialogues, mathematics and reality are considered to be equivalent. The pure mathematician, and ultimately the dialectician, are privileged to understand causal and structural realities at a fundamental level. As maintained in the Meno, the manner in which this knowledge is derived is recollective. If the mathematician can perceive reality with greater perspicuity than the tradesman, the soldier, or the craftsman, it is merely because the former is gifted with a greater capacity to recollect what the forms have already disclosed to him (Plato 633-634). It is precisely this capacity for recollection that makes mathematics a worthwhile study for all philosophers and politicians.
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.
Einstein and Wittgenstein provide an alternative to mathematical platonism that could be deployed to deflate traditional platonism and establish a foothold for a prominent anthropocentric ontology of mathematics in the research literature. A generalization of the argument would serve well in discussions that seek to go “between the horns” of Platonistic and Aristotelian discourse, and a sketch is provided here (Aristotle 14-15). Traditional mathematical platonism supposes that abstract mathematical entities (especially sets, functions, and relations) exist independently of human languages and cultural practices. Let us suppose that L is a formal language consisting of a finite collection of symbols with specified semantics and well-formed formulae. If the Law of non-contradiction holds, L cannot both exist and fail to exist. Assume that elements belonging to a set that exists must be granted ontological status – a reasonable assumption that any flavor of Platonist or Aristotelian would likely readily grant. Suppose that there exists a proof that if L did not exist (e.g., the quantities that L references are ontologically prior to L), then the set containing nothing (e.g. the empty set) would still exist. If L does not exist, then by modus ponens, the empty set still exists. Hence, by non-contradiction, L is not an element of the empty set. Hence, L exists, which is a contradiction. A similarly adapted and suitably formalized proof could be used to establish the undecidability of the pre-linguistic existence of the integers, or – in yet more general cases – mereological constructs formed from any suitably restrictive formal language L. The argument could be extended to a proof that sets could not be proven to exist prior to the existence of a formal language L. Because descriptive sets involve quantification over the integers or systems of numeration derived from the integers, the class of descriptive sets could not be taken to exist independently of the integers. If there exists a proof that, in principle, one could never prove or disprove the pre-linguistic ontological status of the integers, then the same conclusion follows for sets quantified over the integers.
An Einsteinian and Wittgensteinian thesis could be used to deflate the pre-linguistic, a-priori existence of general sets. Let us suppose that a descriptive set S consists of brackets and a non-contradictory mereological condition p(s) quantified over a formal language L. If the condition p(s) is of the form “f(s)~k” for some relation ~, well-formed formula f and bound variable k, then the mereology of S requires an interpretation of whether each element r in S meets the condition f(r)~k. This is equivalent to the question of whether f(r) and k share a common property. But the determination of this property requires an interpretation in the language L, and is a series of (possibly contradictory) well-formed L-conditions. If the L-conditions are contradictory (e.g. the assertion that a given integer n is both even and odd at the same time), then by the law of non-contradiction, the set S must be empty. If the L-conditions are non-contradictory, then S must be non-empty. But if L does not have ontological status, then no L-conditions could exist, and S is consequently empty, which is a contradiction.
This thesis demands that if L is not granted ontological status, each set S whose mereology is founded in well-formed L-conditions must be empty. Consider, however, the existential quantifications that could exist in such a set. Any set that could assert an existential claim would produce a self-contradiction, negating its own existence and forcing all such sets to be equal to the empty set. This set theory deriving from Wittgenstein and Einsteinian model semantics is bizarrely different from a naive set theory, and illustrates several difficulties with a naturalist approach to set theory. In particular, the most intractable contradictions seem very simple to state, but very difficult to dispense with in the absence of a detailed axiomatic strategy. Unfortunately, the same critique can be held to apply to derivative structures as well. Suppose X and Y are sets and that f is a function from X into Y. In light of the foregoing critique, if either X and Y contain an existential quantification, then the function is trivial. If X or Y may fail to possess ontological status at all, then it is undecidable whether f should be granted ontological status either. Functional logic and n-th order quantifications are capable of the same nominalization.
A paradox inherent in the common Einstein and Wittgenstein semantics is the following. Suppose that we must grant ontological status to mathematical entities that appear in well-supported scientific theories of reality. If T is such a theory, and if T could be taken to exist independently of any human language L, then there must be a set of mathematical quantities (constants of nature, physical constants, proportionality constants) that are indispensable to T. But these constants are numbers which, by the above critique, cannot be proven to exist independently of any human language L. If there were to exist a proof that these mathematical objects necessarily existed independently of any language L, there would exist a proof that these numbers existed pre-linguistically. This is a contradiction; consequently, no such proof can exist.
These observations – however halting, tentative, and incomplete – can be interpreted as a partial vindication of Wittgensteinian and Einsteinian philosophies of mathematical language as indispensable to mathematical reality. The concept of mathesis is partially the ontological realization that at least some abstract mathematical theory has its origin in purposive generalizing tendencies of mathematical research programs that seek to make themselves maximally available for descriptive phenomenology. A promising avenue of inquiry – and one which, regretfully, cannot be discussed at length even in future submissions – is the introduction of Poincare’s structuralism to the argument. A case against the primacy of set theory in mathematical discourse could be turned into a search for fundamental structures that are pre-linguistic and contextualize the language of the relevant set theory. The resulting linguistic theory could attempt to evade the undecidability criteria postulated in the current essay by introducing, for each formal language L, a meta-language L’ containing all of the mathematical claims quantified over L. The resulting semantics of L’ could be studied as a sort of Platonistic structuralism while avoiding strong claims of the pre-linguistic nature of mathematical entities.
Hutchins R. M. & Encyclopaedia Britannica Inc. (1955, 1952).
Great books of the western world ([Private library ed.]). Encyclopædia Britannica.