Author: Dr. Jonathan Kenigson, FRSA
The current series of articles 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 begin 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. We shall derive a small but novel result combining four of the ‘best’ current scientific theories: f(R) gravity, String Theory, General Relativity, and related topics in Algebraic Topology – in an effort to show that each can be credibly considered the subject of intuitive generalization rather than abstract objects as in QPT. 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. However, we shall argue that abstract objects should be posited as explanatory for the success of a physical theory if and only if no anthropocentric process like “mathesis” can be imputed to lie behind the success.
A classical tacit epistemological assumption is that mathematics is the artifact of its discourse – proofs, theorems, lemmas, 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 dissertation 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 philosophers of mathematics to equate mathematics with its artifacts (proofs), so that the epistemology of mathematics is to a great degree subordinated to proof per se. The dissertation concludes with a concrete, practical demonstration of the manner in which “mathesis” operates to produce novel results that have predictive physical power. The application field of choice is black hole dynamics – a field whose related theories is well represented in the literature regarding the QPT. These forays collectively represent a demonstration of how “mathesis” operates across broad domains of mathematical research and can supplant the abstract objects of QPT as 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 (70); the Decline and Fall of the Roman Empire also explores the foundations of geometry (Gibbon 299). The purpose of this series of essays is to explore Einsteinian and Wittgensteinian epistemology with a view toward dissolving some intractable Platonistic critiques that appear in discussions of mathematical entities. The current discussion will also explore the historical development of Aristotelian and platonic trends in mathematical thought.
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.
Works Cited.
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. Vol. 48. William Benton, Publisher: Encyclopaedia Britannica, 1952.
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