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First Discourse Upon a Phenomenology of Mathematical Progress Suitably Adapted for Interested Readers of Silicon Valley Executive Network (SVEN) – Silicon Valley, California, USA

Conveyed First to the Bulgarian Academy of Sciences, May 2019

 Approved for Publication by the Academic Senate of the Bulgarian Academy of Sciences, Bulgaria, Sofia, 1040
1 “15 Noemvri” Str.

Dr. Jonathan Kenigson, FRSA

The Western philosophical tradition tends to relegate mathematics either to dialectic or to the applied sciences. Of the first bent, one counts Plato in the Euthydemus (75); Republic Books 6 and 7 (386-388; 391-398); in passing in the Philebus (633; 635); and Book 7 of the Laws (728; 730). The tacit assumption of the Platonic corpus is that pure mathematics is a resource to be exploited for its utility in facilitating clear reasoning as opposed to an end in-itself. This trend paradoxically gives rise to the Plotinian rejection of the primacy of geometry in philosophical discourse as well as the geometric foundationalism of Hobbes in Part 1 of the Leviathan, of which Spinoza’s transcendental rationalism is a refutation (Spinoza 370). Bacon’s assertion in the Foundations of Learning that pure mathematics finds its ultimate utility in its ability to clarify assumptions and deductions is also derivative of Platonism (16 – also, Novum Organon Book 1, APH 80, 120). Conversely, Aristotle maintains that mathematics is distinct from philosophy in its methodology. Physical and metaphysical questions – including those of syllogism and logic – can be explored by means mathematics but not reduced to mathematics per se. This line of reasoning is clearly present in the Metaphysics, Books 1-4 and Book 7, Chapter 8 (270-271; 371; 510; 513; 547-548; 589-593) but also appears in Physics, Book 2, Chapter 2 (270-271) and Book 1 of On the Heavens (371); On the Soul (632); and Nicomachean Ethics (391). Kant attempts a unification of the Platonic and Aristotelian views in Pure Reason (15-17; 19; 211-218); Critique of Judgment (551-552); and Ethics (376). J. S. Mill’s inductive naturalism is a result of the Aristotelian and Kantian metaphysics. Mathematicians have tended to retain a Platonist rhetoric of mathematical discovery but resist Plato’s insistence upon the primacy of dialectic over mathematics.

In a more classical manner, one can locate mathesis in the philosophy of mathematics itself, manifesting as a dialectic between Platonistic and Aristotelian tendencies. This dialectic has never been closely probed by philosophers of mathematics. The effect is a historical amnesia regarding the original motivations of seemingly intractable philosophical problems like indispensability and the nature of abstracta – and perhaps the notion, advanced incompletely by Einstein, Wittgenstein, and Poincare – that no final resolution can ever be made. Indeed, the origin of indispensability arguments is – as is typical of problems in the philosophy of mathematics – the dialectic of classical antiquity. The ontology of a discipline cannot be considered apart from the ontology of its entities, and mathesis represents itself in the alternating Platonistic and Aristotelian tendencies in classical to early-modern philosophy of mathematics, mechanics, and science. The origins and development of mathematics and its several divisions have been deeply studied by classical authors, but there is no consensus regarding the existence of abstract mathematical objects. Among historians, Herodotus (70) and Gibbon (299) mention such studies; and among mathematicians, the accounts of Nicomachus of Gerasa (620), Copernicus (510), Whitehead (126-127; 137–138; 151; 163-164), Poincare (1-5), and Hardy (380-381) are most influential. Among philosophers and historians of ideas, one first counts Plato (254) and Aristotle (119), whose accounts of the nature of mathematical truth roughly parallel the Platonist and formalist accounts, respectively. In their train, one counts Plotinus (558); Augustine (736-737); Hobbes (268); Descartes (228-229); and then the British Empiricists, most notably Locke (362-363) and Hume (458). Hegel’s idealism represents a complete repudiation of the empiricist paradigm (126; 230), whereas Kant attempts to reconcile such accounts in his transcendental analytic of space and number (17-18; 68-69; 553). A prominent subsidiary intellectual trajectory is the demarcation problem – the attempt to define the nature of mathematical knowledge in relation to the knowledge of logic and metaphysics. This inquiry – introduced by Plato (391-398) and Aristotle (270; 391; 547-548) and maturing during the medieval period in the scholasticism of the Summa – advocates that number is actual magnitude and thus existing in the world (238-239; 424-425). This rejection of the eidetic nature of number prompts Hobbes to doubt the essential distinction between mathematics and physics (58; 59; 72; 267), and Descartes to reassert it (227-229; 234-235; 296; 354-355) – a position again repudiated by Kant (5-9; 17-19; 243-248; 311-313). The logicism of Russell (289-290) and the Platonism of Hardy (377-378). Among scientists, the view of mathematics as a ‘language of nature,’ and thus prior to physics and metaphysics, has persisted; one can find this view among the cosmologists Ptolemy (5-6), Galileo (190), and Newton (1-2); the agenda finds its epitome in Pascal’s transcendental rationalism (445) as well as in Hegel’s dialectical rationalism (230).

Mathesis is further represented in the dialectic regarding the nature of the certainty of mathematical knowledge and – in particular – whether arithmetic and geometry enjoy an a-priori foundation. The discussion, it might be said, vacillates ad-infinitum between Platonic and Aristotelian perspectives. Plato addresses the question of mathematical a-priority first in the Philebus (633-634); Aristotle considers it cursorily in the Posterior Analytics (119) and more fully throughout in Metaphysics Book 1, Chapter 2 (500), Book 2, Chapter 3 (513), and Book 13, Chapter 3 (609-610). The Nicomachean Ethics also provides a cursory analytic (339-340). Among mathematicians and scientists, the platonic trajectory is continued in Book 1 of the Arithmetic of Nicomachus (811-814); the Almagest of Ptolemy (5-6); the Two New Sciences of Galileo (252); and the Geometrical Demonstrations of Pascal (430-446). In the Meditations, Pascal briefly considers the apriority of number in reflections 1-2 (171-172) and 33 (176), as do Lavoisier in Chemistry (1-2) and Fourier in the Theory of Heat (170-174). These reflections fall short of explicit philosophical arguments, but are nonetheless reflective of the scientific zeitgeist. Among metaphysicians, the Platonic trajectory is expressed fully in Berkeley’s Human Knowledge (436); the a-priority of number is defended throughout Kant’s corpus, but most explicitly in the Critique of Pure Reason (5-8; 15-16; 31; 35-40; 86; 211-218) and indirectly in the Critique of Practical Reason (295; 312; 330-331); the Science of Right (399); and the Critique of Judgment (551-552). Descartes defends the a-priority of mathematical knowledge in his first meditation (76; 93-95); Objections and Replies (128-129); and the second book of Geometry (304). Of the Cartesian corpus, the Discourse of Method, parts I and II (43-47) as well as parts IV and V (52-55) provide the most familiar defenses of mathematical apriority, although it is incompletely discussed in the Rules (2-7) as well. One finds in Locke a detailed refutation of Cartesian arguments of apriority in Human Understanding Book IV, chapters II and III (311; 317-320; 322-323); chapter IV sections VI-IX (325-326); and very strongly in chapter XII (358-360). Hume’s Human Understanding follows a similarly skeptical course in section IV, division XX (458) as well as section VII (470-471) and section XII (508-509). Mill briefly discusses a refutation of Platonism in Utilitarianism (445) and indirectly in Liberty (283-284), and James considers such in his Psychology (175-176; 874-878; 879-882). 

The dialectic between Platonistic and Aristotelian perspectives represented by mathesis also inform an intimately related discussion concerns the question of whether mathematical entities have an eidetic, physical, or mental existence. Plato’s Phaedo (228-231; 242-243); Republic books VI (387) and VII (391-398); and Theateatus (535-541) adopt an explicitly eidetic view. This perspective is reinforced in passing in the Philebus (636) and more cursorily in the Sophist (562) and the Seventh Letter (809-810). The breadth and depth of the Aristotelian refutation is notable, and is clearest in the Posterior Analytics Book 1, Chapters 10 (105), 13 (108-109), and 18 (111) and the Metaphysics Book 1, Chapters 5-6 (503-504; 505-506); and Chapters 8-9 (508; 509-511). A further concentrated discussion of this subject occurs in Book 3 of the Metaphysics, especially Chapters 1-2 (515-516); Chapters 5-6 (520-521); Book 7, Chapter 2 (550-551); and Book 13, Chapter 3 (606-610). Aristotle’s discussion in Nicomachean Ethics Book 1, Chapter 6 (341) and Book 6, Chapter 8 (390-391) also advances his characteristic objections to Platonism as presented in the Theateatus, and Republic. In the Elements, Euclid reiterates a Platonist view of abstract quantities, especially in Book 1, postulates 1-3 (2) – a view more thoroughly advanced by Nicomachus in his Arithmetic (812; 814). Medieval commentators are predictably scholastic in nature. In Christian Doctrine, Augustine references the innateness of ideas of infinity and imputes to mathematics the symmetry of the divine mind (654), a view mirrored briefly in Book 10 of the Confessions (76-77) among other passages. A similar view is advanced in various passages by Aquinas in his Summa, Q. 1 (25); Q. 10 (45-46); Q. 11 (46-47; 49); as well as Q. 30; Q. 44; and Q. 85 (167-168; 238-239; 451-453), among other brief discussions. Presaging Kant, Part 3 of the Summa further supports the objectivity of mathematical reality and the innateness of mathematical ideas: in particular, Q. 7 (754-755); Q. 83 (976-978; 978-980), among other discussions, including Q. 83, A. 3, Rep. 2. Despite Aquinas’ view that perception is the source of knowledge, mathematics remains inviolably certain. Descartes’ break with scholasticism does not dethrone the ultimate certainty of mathematical knowledge – a view championed in Rules (particularly in XIV, 30-32); Discourse, Part IV (52-53); Meditations (76; 93; 96); and Objections and Replies (169-170; 217-218; 228-229). The transcendental existence of mathematical objects figures in Spinoza’s rationalism as presented in Part 1 of the Ethics (370), as well as Pascal in Treatise on Vacuum (373), despite the latter’s scientific empiricism. Berkeley’s transcendental empiricism defends the novel view that mathematical entities – like physical ones – have an ideal, mental existence that is objective. This view is advanced in Human Knowledge, Sections 12 and 16 (408-409; 415); Sections 118-128 (436-438); and very forcefully in Sections 121, 122, 125, and 126 (436-438). Hume’s characteristic empiricism is expressed in Human Understanding, Section 4, Division 20 (458); Section 12, Division 122 (505); and Division 125 (507). Kant elegantly reconciles Humean empiricism and Cartesian idealism throughout the Critique of Pure Reason (16; 24-33; 35-36; 46; 62; 87; 91-95; 211-212) and in passing in the Critique of Practical Reason (295; 312) and the Critique of Judgment (551-553). In the Analytical Theory of Heat, Fourier briefly defends the ideal existence of mathematical entities and more broadly posits their usefulness in exploring the hidden relationships in nature (183). Pragmatic philosophy, stemming from James’ Principles of Psychology, posits mathematics as a set of mental constructs that are useful in interpreting empirical reality (see, for instance, 874-878; 880; 881; 875-876 is particularly relevant).    

The dialectic of mathesis also proceeds to consider the objects of mathematical knowledge itself, which are often the foci of philosophical inquiry in modern times. Plato makes such discussion in his exploration of sophrosyne (temperance) in the Charmides (7-8) and in his dialogue on virtue (arete) in the Meno (176-177; 180-183), marking arete itself a mathematical expression. Book VII of the Philebus (633) and books VI (387) and VII (393) of Republic conclude the platonic discussion. The objects of mathematics are taken to be geometric and arithmetic quantity – number, figure, and relation. Later authors – particularly mathematicians – hold variants of this view; for instance, Newton in his Principles (1), and Galileo (252-254). For Aristotle, however, the objects of mathematics are abstractions, and the objects of mathematics are thus found both in reason and perception. This is the view taken in Physics Book IV (299); Book III, Chapters I (514) and V of the Metaphysics (520-521). Further argument along these lines can be found in Book VII, Chapter II (551) and Book XI, Chapter XIII (589) in the Metaphysics; Categories, Chapter 6 (9-11); and Categories Chapter 8 (15), among other locations. Interestingly, the Rhetoric (595) briefly discusses the objects of mathematical knowledge, showing the ubiquity of Aristotle’s interest in this knowledge among the liberal arts. Augustine takes this trajectory in his Christian Doctrine, Book 2, Chapter 38 (654) and Aquinas in the Summa Part 1 (32-34; 167-168; 451-453). In Books I and VII of the Elements (1-2; 127), Euclid makes a careful enumeration of the objects of geometric proof, concluding with the definitions of number theory in Book XI (301). A derivative effort in this regard may be found in Ptolemy’s Almagest, Book 1. In contrast to these geometers, Nicomachus develops the objects of arithmetic more fully, taking Book XI of Euclid as a starting point and providing a detailed taxonomy of number in his Arithmetic (811-812). Descartes’ universal mathesis seeks a fusion of Nicomachean and Aquinian perspectives in analytic geometry, resulting in the classification of number and figure as similar abstracta in Rules (3; 7; 8-10; 30-33); the Discourse Parts II and IV (47; 52-53); Meditations (76); and Book II of the Geometry (304-306; 316). The empiricists do not rebel strongly against the Cartesian revolution in this regard, as can be seen in Book II, Chapter XII of Locke’s Human Understanding (147-148); Section 118 of Berkeley’s Human Knowledge (436; 439); and Hume’s Human Understanding (508-509). In the Critique of Pure Reason, Kant establishes the objects of mathematics as faculties of abstraction that exist innately in the human mind, per 46; 62; 68-69; and 211-213. This pseudo-Aristotelian Psychologism is brought to the fore by James in his Psychology (874-878). A developed philosophical interest in the application of mathematics in physics is also notable. Platonic perspectives can be gleaned from books VII-IX of the Republic (391-398; 403; 424-425); in the Timaeus (448-450; 452-454); in the Philebus (633); and in Books V and VII of the Laws (691-692; 695-697; 728-730). While mathematics is conceived as useful in physics, the study of mathematics should, for Plato, be pursued for the sake of its applicability in metaphysics (see, for instance, the Republic above).

Works Consulted.

Pagination has been updated to reflect references from the 1952 edition of the Great Books of the Western World, in the public domain thanks to the generosity of the Library of India. 

Adler, Mortimer Jerome. The great ideas: A syntopicon of great books of the western world. Vol. 1. Encyclopaedia Britannica, 1952.

Adler, Mortimer Jerome. The great ideas: A syntopicon of great books of the western world. Vol. 2. Encyclopaedia Britannica, 1952.

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