1 Realists and Nominalists: Language and Mathematics before the Scientific Revolution
While much of medieval and Renaissance grammar, logic, and metaphysics was based on interpretations of classical texts, it would be a mistake to assume that the texts of Plato and Aristotle that were read in the thirteenth, fourteenth and fifteenth centuries were equivalent to those we recognize as genuine today.1 A number of spurious texts were included among the canonical works of Aristotle, and discrepancies in translations and manuscript editions of texts often resulted in variations among editions of the same work. Furthermore, as I will discuss in the next section, the ‘Aristotelian’ philosophy of scholastic authors such as Thomas Aquinas should not be equated with the philosophy of Aristotle himself, since medieval authors’ interpretations drew on a wide variety of sources, and in many cases went to great lengths to reconcile Greek philosophy with Christian doctrine, often at the expense of fidelity to the original texts. This is particularly an issue when confronting sixteenth- and seventeenth-century critiques of ‘Aristotelian’ philosophy, which are often directed more at scholastic interpretations than at the actual teachings of Aristotle.
It is still true, however, that most of the central issues in episte-mology over the past two millennia received their first systematic treatment by either Plato or Aristotle, and that later authors—both realist and nominalist—consciously built on these foundations. The positions produced in classical antiquity that we are concerned with can be grouped roughly as follows: (1) the actual teachings of Plato and Aristotle; (2) independent mathematical works by technical authors such as Apollonius, Euclid, Archimedes, and others; (3) commentaries on either Plato or Aristotle, such as Proclus’ highly influential Platonic reading of Euclid; and (4) reactions against Plato and/or Aristotle, including those of the Epicureans, Stoics, and the Academic and Pyrrhonian skeptical schools. This grouping reflects those works or philosophical movements that were most directly influential during the Renaissance and Scientific Revolution, and my sketch here is mostly for the purpose of contextualizing the later portions of this study. It is not intended as a comprehensive survey of classical epistemology.2
The Philosophy of Language and Mathematics in Antiquity
Greek authors’ interest in epistemological questions about language originated at roughly the same time as early speculations about cosmology and mathematics, most notably in the writings of the pre-Socratic authors Parmenides, Protagoras, and Prodicus. This early study was mostly concerned with the origin and etymology of language, but certain authors, such as Parmenides, actually developed a more comprehensive theory of signification, centered on the proper application and interpretation of ‘names.’3 This theory, which was later modified by Protagoras and Prodicus, assumed a correspondence between names and objective reality, and thus may be seen as a forerunner to Plato’s idealistic treatment of language. Plato takes up this issue in the dialogue Cratylus , where he contrasts the notion that language has arbitrary conventional signification with the belief that words themselves somehow embody the essence of their referents through similitude or onomatopoeia. Socrates dismisses the latter notion, concluding that the proper origins of knowledge are to be found not in the formal characteristics of words themselves but rather in the ideas they signify. The word ‘horse,’ for example, does not signify a particular object because of any necessary relationship between its arrangement of letters or the sound produced when spoken aloud, but rather because it calls to mind a particular idea which directly corresponds to the animal.4
Like Plato, Aristotle was interested in language primarily as the vehicle for thought, and viewed mental states as symbols to which words refer. For both Plato and Aristotle, the conventional aspect of particular languages was not important; rather, the intriguing question was the epistemic basis for connecting words with things, a link which both authors concluded depended upon the reliability of ideas to simultaneously act as referents of words and to signify real objects. Aristotle sums up this chain of signification in On Interpretation, where he distinguishes further between written and spoken language:
spoken words are the symbols of mental experience and written words are the symbols of spoken words... The mental symbols, which these [words] directly symbolize, are the same for all, as also are those things of which our experiences are the images.5
In claiming that the ‘‘mental symbols’’ that words signify ‘‘are the same for all,’’ Aristotle distinguishes between the conventional aspect of ‘a language’ and the universal function of language in general by conceding that while individual words may be employed conventionally, the mental images they signify are not. Because mental pictures are derived from the experience of ‘‘things,’’ and because we all have the same mental pictures of things, Aristotle suggests there must be either an inherent correspondence between all minds and the world, or else all minds must contain an innate set of categories shared by everyone. Aristotle also maintains that while one function of words and language is to facilitate communication, logical determinations (such as causal relationships) are not based in language but rather are merely reflected in it—knowledge, in other words, is not ‘in language,’ but rather ‘in the mind.’6 As he states in the Posterior Analytics:
all syllogism, and therefore all a fortiori demonstration, is addressed not to the spoken word, but to the discourse within the soul, and though we can always raise objections to the spoken word, to the inward discourse we cannot always object.7
This kind of realism about the connection between language, ideas, and the world was the standard position taken in antiquity, and it informed most of the linguistic theories of the Middle Ages and the Renaissance. Other ancient commentators, such as the Stoics Zeno and Diogenes Laertius, believed that the mind was formed to reflect the rational order in nature, which privileged mental imagery in much the same way as Plato’s doctrine of forms. It should be noted, however, that at least one significant strain of Greek philosophy rejected this position in favor of a more rigorous conventionalism: Aristotle reports that Democritus advocated a conventional theory of language, and Epicurus’s Letter to Herodotus promotes a conventional attitude towards language coupled with a quasi-mechanical theory of sense perception and idea formation that is explicitly anti-mentalistic.8 These examples become significant when we see later that the seventeenth-century nominalist movement was partially inspired by these classical atomist authors.9
On the subject of mathematics as a mode of representation, Plato and Aristotle promote views consistent with their linguistic theories. Plato is famous for his declaration that all students in his Academy should be versed in geometry, but there is little evidence that he himself was an accomplished practicing mathematician. His ontology, however, is explicitly geometrical, and his dialogue Timaeus is probably the most frequently cited source for the belief that the universe is a divine geometrical construction. Plato states here that God ‘‘judged that order was in every way better’’ than disorder and established a structure based on the perfection of the regular geometrical solids.10 This geometrical cosmology is backed by an assumption that mathematics, being a science of the ‘eternal and unchangeable,’ represents the most secure knowledge and is the highest standard of truth, and Plato’s epistemology consequently prioritizes reason above the senses.11 This doctrine relates to his theory of forms, which posits that our knowledge of mathematics is underwritten by the intellect’s perception of archetypal figures that are eternal and exist outside of the realm of the senses. As a general epistemological program, this extreme idealism had only limited influence on later philosophical movements. Aristotle rejected his teacher’s doctrine of forms, and while it enjoyed a revival among fifteenth and sixteenth-century Neoplatonists and seventeenth-century Platonists, Platonic idealism did not agree particularly well with the experimental spirit of the Scientific Revolution. Specifically as a philosophy of mathematics, however, Plato’s arguments were enormously influential: Kepler, Galileo, Descartes, Leibniz and many other founders of the ‘modern’ mathematical method could variously be described as having Platonic conceptions of mathematical concepts and entities, and in the twentieth century there are yet mathematicians who would willingly call themselves ‘Platonists.’
Plato’s philosophy of mathematics is fully integrated into his general theory of knowledge, and in a number of dialogues he explicitly uses mathematical examples to illustrate his epistemological beliefs.12 Plato was a metaphysical realist, and held that when we perceive individual objects, we are accessing not their particular qualities but rather their essential nature or the real universal category they fall into. To illustrate this, Socrates presses Theaetetus to describe how universal notions are formed from particular objects, and particularly wonders ‘‘through what bodily organ the soul perceives odd and even numbers and other arithmetical conceptions.’’ Theaetetus responds—to Socrates’ approval—that his ‘‘only notion is, that these [mathematical objects], unlike objects of sense, have no separate organ, but that the mind, by a power of her own, contemplates the universals in all things.’’13 This aspect of Plato’s epistemology relates directly to his theory of ‘recollection' or reminiscence, which is developed primarily in Meno and Phaedo, and becomes a paradigm for human apprehension of geometrical ideas. In order to demonstrate to Meno that we are born with the knowledge of essences built into our minds, Socrates questions an unschooled servant about the construction of a geometrical figure. Although he has never learned geometry, the servant is able to answer Socrates' questions, which causes Socrates to conclude that ‘‘he who does not know may still have true notions of what he does not know.'' Socrates explains that the servant is able to ‘recover' knowledge of ‘‘geometry and every other branch of knowledge,'' since ‘‘there have always been true insights in him, both at the time when he was not a man, which only need to be awakened into knowledge by putting questions to him.''14
Socrates uses the discussion of recollection as proof of the immortality of the soul, since the objects of truth, which are immaterial and eternal, can only be apprehended by something that is also eternal. It is from this notion that Plato develops his strong dichotomy between sensory (corrupt) experience and rational (perfect) knowledge, and while he applies this model to many kinds of concepts—‘truth,' ‘beauty,' ‘the good'—it is particularly well suited to mathematics. In Plato's account, mathematical objects are real objects in the purest sense: they are totally disembodied from physical form, and their existence is not inferred or abstracted from crude matter, but rather is present to the intellect prior to any experience of physical nature. Because of this Plato is ambivalent about the use of language—the use of particular words to signify concepts in a particular language, that is—but not about the ability of the mental images signified by words to accurately represent true objects of knowledge. In a sense, an idea of an essential nature actually is that object, since it is perceived by the soul, which is made of the same ‘substance’ as the ethereal substratum occupied by the forms.
This is one major difference between Plato's and Aristotle's philosophies of mathematics: Aristotle's approach is integrated into his overall logical method, and therefore is developed more distinctly and explicitly than Plato's. As Deborah Mondiak notes, Aristotle's philosophy of linguistic meaning (as expressed in De Interpretatione ) informs and is supported by his more epistemological works (such as Posterior Analytics , Physics , and Metaphysics), and she argues that ‘‘Aristotle’s intent [in De Interpretatione]... is to give an account of language and its relation to the world that supports, inter alia , the realist epistemology of the Posterior Analytics.''15 Like Plato, Aristotle was also knowledgeable about the mathematical developments of his day, and he followed Plato in often using mathematical examples to illustrate metaphysical and logical arguments.16 In a sense, Aristotle’s metaphysics is reducible to logic: it is concerned with categories of existence and with the classification of objects according to these categories. Unlike Plato, Aristotle denies primary existence to abstract entities, which is part of his rejection of Plato’s doctrine of forms.17
One consequence of this position is that metaphysical questions concerning existence require logical definitions. Since Aristotle’s metaphysics holds logical consistency as its highest criterion for truth, one might wonder whether Aristotle’s notion of existence is simply a logical one.18 In the Posterior Analytics , Aristotle addresses the problem of definitions, noting that a definition cannot confer knowledge of the essence of an object. Using the example of the notion ‘triangle,' Aristotle explains that neither the word nor its associated definition reveals the ‘nature’ of the geometrical object. Aristotle points out that while a definition may be true in the sense that it does not violate any logical criterion for existence, this ‘truth' does not prove that the object defined necessarily must exist, and he concludes that ‘‘definition neither demonstrates nor proves anything, and that knowledge of essential nature is not to be obtained either by definition or by demonstration.''19 This does not, however, mean that Aristotle believes that essences either do not exist or are not accessible by reason. Aristotle defines ‘‘essential nature’’ instead as ‘‘the cause of a thing’s existence,’’ and he notes that a syllogistic demonstration (as opposed merely to a definition) which logically deduces the cause of a thing is capable of revealing the thing’s essence.20
The question then is whether mathematics, which, according to Aristotle, is a logical science, has as its objects real objects whose essences can be causally known, or rather is merely a set of definitions. This discussion takes place primarily in the Metaphysics, and the conclusions are somewhat equivocal. Aristotle does state categorically that mathematical objects do have some kind of existence, but suggests that the interesting question is ‘‘not whether they exist but how they exist.’’21 He begins by surveying the popular opinions on this subject, going into a lengthy and critical discussion of Plato’s belief that ‘‘the Ideas are substances,’’ which he eventually dismisses. A crucial question for Aristotle is whether mathematical objects exist as ‘primary’ or ‘secondary’ substances—that is, whether they exist as separate entities or rather as accidents tied to perceptible objects. This discussion has particular relevance to arithmetic, since a central question concerns the status of number or ‘‘discrete quantity,’’ as opposed to continuous geometrical quantity. Aristotle concludes that, since every instance of discrete quantity is related to a perception of some other object—‘three horses,’ ‘five stones,’ etc.—numbers are accidents, existing only in relation to some physical substance. He bases this reasoning on the principle that two primary objects cannot occupy the same space; supposing numbers to exist as separate entities would necessitate the simultaneous existence of both the physical object (‘stone’) and its quantity (‘one’), which is a physical impossibility.22
Granting mathematical objects existence as accidents, however, circumvents this physical difficulty, since it allows for quantity to be ‘in' physical objects without contradiction, and still preserves mathematical objects as ‘abstract entities' with a kind of real existence. Aristotle concludes
Thus since it is true to say without qualification that not only things which are separable but also things which are inseparable exist, it is true also to say without qualification that the objects of mathematics exist, and with the character ascribed to them by mathematicians.23
Elsewhere Aristotle argues for the primacy of arithmetic over geometry, since his int...
