Mathematics

Plato assigns to mathematics both a high dignity and a definite limitation. In the Republic, he ranks the mathematical sciences, arithmetic, geometry, astronomy, and harmonics, above opinion and above the study of visible things. The mathematician grasps objects that are real, stable, and intelligible. The triangle the geometer reasons about is not the figure drawn on sand or papyrus; it is a Form, apprehended by the intellect alone. Mathematical objects do not come into being or pass away. They do not depend on anyone's perception. As arithmetic deals with numbers and not with numbered things, so geometry deals with figures and not with physical shapes.

Yet Plato holds that mathematics is not the highest knowledge. In the Divided Line, mathematical reasoning (dianoia) occupies the second-highest segment, below dialectic (noesis). The mathematicians begin from hypotheses, from definitions and axioms they take as given, and reason downward to conclusions. They do not turn back to examine those starting points. "These are their hypotheses, which they and everybody are supposed to know, and therefore they do not deign to give any account of them." The dialectician, by contrast, ascends from hypotheses to the unhypothetical first principle, the Form of the Good, and sees all things in its light. The relation of mathematical knowledge to dialectic is treated more fully under the idea of Dialectic.

In the Meno, Socrates leads a slave boy through a geometric proof, and the boy arrives at correct conclusions about doubling the area of a square without having been taught. Plato's point is that the boy is not learning something new from experience but recollecting what his soul already knew. Mathematics thus provides evidence that the mind possesses knowledge independent of the senses, and it is for this reason that Plato recommends it so highly as preparation for philosophical inquiry.

Plato's ranking of mathematics below dialectic raises a question that recurs throughout the tradition: whether mathematical certainty can be explained without positing a separate realm of intelligible objects. Aristotle will argue that it can, by treating mathematical objects as abstractions from physical things; Kant will contend that the certainty belongs to the mind's own structure rather than to objects in any realm at all.

Aristotle rejects Plato's view that mathematical objects exist in a separate, intelligible realm. Numbers and figures are real, but they are not substances. They are properties of physical things that the mind isolates by a process of abstraction. When the geometer studies a sphere, he studies the spherical shape belonging to bronze balls and water drops, considered apart from their matter. The sphere does not exist on its own, free of all material instances. It exists in things, and the mathematician thinks about it as though it were separate, without claiming that it actually is. "Mathematical objects," Aristotle writes, "exist neither as separate from material things nor as in material things."

This distinction has consequences for Aristotle's account of the sciences. Each science studies a particular domain by abstracting relevant features from the full complexity of nature. Physics studies bodies in motion. Mathematics studies quantity and spatial figure. Metaphysics studies being as such. The mathematician does not deal with a higher reality than the physicist; he deals with the same reality under a more abstract description. Mathematical precision is possible because mathematics strips away the features, such as change, matter, and purpose, that make physical inquiry difficult. Yet Aristotle complains of those who suppose "the minute accuracy of mathematics is to be demanded in all cases," for its method is not that of natural science.

In the Posterior Analytics, Aristotle develops his theory of demonstration, which owes much to mathematical practice. A demonstration is a syllogism from premises that are true, primary, and better known than the conclusion. Mathematics exemplifies this structure, and Aristotle draws many of his logical examples from geometry. Yet he insists that mathematical proof is one species of demonstration, not the model for all reasoning. The physicist and the ethicist reason differently, and their subjects demand it. The relation of mathematical method to other forms of demonstration is considered further under the idea of Logic.

Aristotle's abstractionism leaves open the question of how mathematical truths about objects that have never been physically realized, such as perfect circles or infinitely large primes, can carry the certainty they seem to possess. Euclid's demonstrative practice will sharpen this question; Descartes will abandon abstraction from nature altogether and propose mathematics as the template for all knowledge.

Euclid's Elements is not a philosophical treatise, yet it bears upon philosophical questions about the nature of mathematical knowledge more directly than many works of philosophy. Plato and Aristotle debated the status of mathematical objects; Euclid, without entering that debate, showed what mathematical knowledge looks like when carried out with full rigor. He begins with twenty-three definitions (a point is that which has no part; a line is breadthless length), five postulates (a straight line can be drawn from any point to any point), and five common notions (things equal to the same thing are equal to one another). From these principles, he derives 465 propositions, each proved by logical deduction from what has already been established.

The basic principles, as Euclid expounds the science, are threefold: definitions, postulates, and axioms or common notions. The axioms are common to other branches of mathematics as well as to geometry, and their truth is supposed to be self-evident. The postulates, by contrast, are peculiar to geometry, for they are written as rules of construction. They demand that certain operations be assumed possible, such as the drawing of a straight line or a circle. When, in his first proposition, Euclid demonstrates the construction of an equilateral triangle, he has in effect proved the geometrical existence of that figure within the space his postulates determine. Until such a construction is demonstrated, a definition states only a possibility to which no geometrical reality is known to correspond.

Book V presents a theory of proportion, attributed to Eudoxus, that handles both commensurable and incommensurable magnitudes. The discovery that the diagonal of a square is incommensurable with its side had raised a difficulty for Greek mathematics, and the Eudoxan theory, as Euclid presents it, addresses the difficulty with great care. The treatment anticipates, as nineteenth-century mathematicians recognized, certain features of the modern theory of real numbers. The question of commensurable and incommensurable quantities is discussed further under the idea of Quantity.

Euclid's method raised a question that later thinkers could not avoid: whether philosophy can achieve the same certainty by proceeding from similarly evident axioms. Descartes will attempt this, and Spinoza will set forth his ethics in geometrical order. Kant, however, will argue that geometry's certainty depends not on the self-evidence of Euclid's postulates but on the mind's own form of spatial intuition.

Aquinas follows Aristotle in holding that mathematical objects are abstracted from sensible things, but he situates this claim within a broader classification of the sciences according to degrees of abstraction. Physics, or natural philosophy, studies things that exist in matter and cannot be understood without reference to matter. Mathematics studies things that exist in matter but can be understood without it: numbers, figures, ratios. Metaphysics studies things that can both exist and be understood apart from matter: being, substance, cause. "The mathematical," Aquinas writes, "do not subsist as separate beings," but "have a separate existence only in the reason, in so far as they are abstracted from motion and matter."

The certainty of mathematics, for Aquinas, follows from the nature of its objects. Because mathematical objects are abstracted from change and individual matter, they are stable, uniform, and accessible to the intellect in a way that physical objects, subject to contingency and variation, are not. A triangle does not grow, decay, or act unpredictably. Its properties can be known with a firmness that arguments in ethics or politics rarely achieve. Yet Aquinas does not conclude from this that mathematics is the highest science. Sacred doctrine proceeds from principles known by the light of a higher science, namely God's own knowledge, and metaphysics addresses questions of purpose, goodness, and ultimate causation that lie beyond the scope of quantity.

Aquinas also observes that different sciences may prove the same conclusion by different means. The astronomer proves that the earth is round by mathematics; the natural philosopher proves the same by considering the nature of matter. This distinction between the formal and the material approach to the same subject bears upon the later question of how mathematics relates to physics, a question treated more fully under the idea of Physics.

Aquinas's hierarchy of abstraction raises a question that the scientific revolution would press with new urgency: if mathematics is certain because it abstracts away from matter, what happens when mathematics is applied back to matter to produce a new physics? Descartes will argue that matter is essentially extension, thus collapsing the distinction between mathematics and the physical world that Aquinas had taken for granted.

Descartes represents a return to the Platonic estimate of the importance of mathematics for the rest of philosophy, though on different grounds. For the ancients and medievals, mathematics was one science among several, distinguished by its degree of abstraction. For Descartes, mathematics becomes the pattern that all genuine knowledge must follow. In the Rules for the Direction of the Mind, he argues that only two operations of the intellect yield certainty: intuition, the immediate grasp of a simple, self-evident truth, and deduction, the necessary inference from one such truth to another. These are the operations at work in arithmetic and geometry, and Descartes proposes to extend them to every subject. "Of all those who have hitherto sought truth in the sciences," he writes, "mathematicians alone have been able to find any demonstrations, any certain and evident reasons."

The practical fruit of this ambition is analytic geometry. By identifying points with ordered pairs of numbers and curves with algebraic equations, Descartes unifies two disciplines that the ancients had kept separate. Geometry becomes a branch of algebra; spatial relations become numerical relations. In Descartes' own view, this tended to unify all existing branches of mathematics and to form a single universal method of analysis. The Cartesian synthesis of algebra and geometry violates the ancient distinction between continuous and discontinuous quantities, between magnitudes (like lines and planes) and multitudes (or numbers), and in doing so it opens a profound discontinuity between modern and ancient mathematics.

The Discourse on Method generalizes these mathematical convictions into four rules for inquiry: accept nothing as true unless clearly and distinctly perceived; divide problems into their smallest parts; proceed from simple to complex; review thoroughly to omit nothing. Descartes believed that if philosophers would reason as geometers do, they could achieve in metaphysics and physics the same certainty that Euclid achieved in geometry. The question whether such certainty is attainable outside mathematics is examined under the idea of Knowledge.

Descartes' analytic geometry opened the path to the calculus of Newton and Leibniz, and his broader ambition, to make all knowledge as certain as geometry, set the terms for much of modern philosophy. Kant, however, will argue that the philosopher who tries to follow the method of mathematics in his own inquiries is led astray by the very brilliance of mathematical example.

Hobbes gives mathematics a nominalist interpretation that differs sharply from both the Platonic and the Aristotelian accounts. For Plato, mathematical objects are eternal Forms; for Aristotle, they are properties abstracted from things; for Hobbes, they are consequences of how we use names. Reasoning, on this view, is computation: "adding" and "subtracting" names according to definitions. When we say that two and three make five, we are asserting that the name "five" is equivalent to the name "two and three." Mathematics is certain not because it touches a higher reality but because we ourselves laid down the definitions from which it proceeds.

Hobbes draws a further consequence from this position. Geometry, he argues, is demonstrable because "the lines and figures from which we reason are drawn and described by ourselves." In geometry, we begin from definitions we construct, and we demonstrate properties that follow necessarily from those constructions. Because we made the objects, we can know them completely. In natural philosophy, by contrast, we study things we did not make, and we cannot penetrate to their inner constitution. The maker has a privileged access to what he has made, and in geometry we are the makers. This same principle, Hobbes contends, extends to civil philosophy, since we make the commonwealth ourselves. The questions raised by this analogy between mathematical and political knowledge are considered further under the idea of State.

Hobbes took this principle seriously enough to attempt his own geometric demonstrations, including several purported squarings of the circle. These efforts were challenged by John Wallis, the Savilian Professor of Geometry at Oxford, who found them fallacious. The quarrel between Hobbes and Wallis lasted decades. Yet Hobbes's philosophical contention, that mathematical certainty derives from the conventional character of definitions, survived his geometric errors and appeared again in later conventionalist accounts of mathematics.

Hobbes's nominalism leaves open the question of why definitions we chose should so reliably describe a world we did not make. Kant will address this puzzle from a different direction, arguing that mathematics describes nature necessarily because the mind's own constructive activity shapes the conditions under which nature can appear to us.

Locke holds that mathematical knowledge is genuinely certain, and he grounds that certainty in the nature of our ideas rather than in the nature of external objects. Knowledge, for Locke, is the perception of the agreement or disagreement of ideas. In mathematics, the ideas in question, numbers, figures, and their relations, are clear, distinct, and determinate. When we perceive that the three angles of a triangle equal two right angles, we perceive a necessary connection between our ideas. This perception may be either intuitive (immediate) or demonstrative (reached through a chain of intermediate ideas), but in either case it is certain. "The knowledge we have of mathematical truths," Locke writes, "is not only certain, but real knowledge; and not the bare empty vision of vain, insignificant chimeras of the brain."

This account allows Locke to agree with Descartes on the certainty of mathematics while rejecting the doctrine of innate ideas. We are not born knowing that two plus two equals four. We acquire our ideas of number through experience. But once we have those ideas, we can perceive their relations with a clarity that experience itself never provides. Mathematical knowledge is thus a priori in its justification, even though its materials come from sense and reflection. For Locke, mathematics is strictly a science of the relations between ideas, not of real existences, a position that James will later reaffirm.

Locke distinguishes mathematical knowledge from our knowledge of the physical world. In physics, we rely on sensation, and sensation gives us only probable belief, not certainty. We observe regularities but cannot perceive the necessary connections between qualities of bodies. Mathematics escapes this limitation because its objects are our own ideas, fully transparent to the mind. The cost of this certainty is a narrowed scope: mathematics tells us about relations among ideas, not about the ultimate constitution of things. The distinction between demonstrative and probable knowledge is treated more fully under the idea of Knowledge.

Locke's treatment of mathematical certainty raised a question the empiricist tradition would continue to press: how knowledge can be both empirical in origin and certain in character. Hume will push the position further, and Kant will take the problem as a starting point for a new account of mathematical truth as synthetic a priori.

Kant insists, even more than Aristotle, that the philosopher is misled if he tries to follow the method of mathematics in his own inquiries. "The science of mathematics," he writes, "presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience. Such examples are always contagious." But the philosopher who expects the same success outside the field of quantities mistakes the source of mathematical certainty. For Kant, the question his predecessors failed to ask clearly is: how is mathematics possible? Mathematical propositions are necessary and universal; seven plus five equals twelve, and no experience could refute it. Yet Kant argues that such propositions are also synthetic: they add content not contained in the bare analysis of concepts. The concept of seven, plus the concept of five, does not by itself contain the concept of twelve. One must perform an act of construction to arrive at the sum.

Kant's solution is that mathematics is grounded in the pure forms of sensible intuition: space and time. Arithmetic depends on the form of time, the successive synthesis of units. Geometry depends on the form of space, the construction of figures in pure spatial intuition. These forms are not features of things in themselves; they are the conditions under which any object can appear to a human mind. Because space and time structure all possible experience, mathematical truths hold necessarily for every object of experience, but they tell us nothing about things as they are apart from our mode of perceiving them. "The exactness of mathematics," Kant holds, "depends on definitions, axioms, and demonstrations," and none of these can be achieved by the philosopher in the sense in which the mathematician understands them.

This account resolves the puzzle that Locke and Hume left open. Mathematical knowledge is not analytic (as Leibniz had suggested), not merely empirical, and not a report on Platonic Forms. It is synthetic a priori: produced by the mind's own activity of constructing concepts in pure intuition, and valid for all experience because experience itself is shaped by the forms that make mathematics possible. The price is that mathematical truth is confined to the realm of appearances; it does not penetrate to the thing in itself. The broader consequences of Kant's doctrine of a priori forms are considered under the idea of Space.

Kant's theory of mathematics as synthetic a priori knowledge has been much debated. Russell rejects the Kantian view that mathematical reasoning always uses intuitions, contending that by the help of symbolic logic all mathematics can be deduced from logical principles alone. Intuitionists such as Brouwer accept Kant's emphasis on mental construction while rejecting his specific account of space and time. The question Kant posed, what makes mathematical truth both necessary and substantive, remains at issue.

Hegel acknowledges the certainty of mathematical knowledge but denies it the status that Descartes, Spinoza, and Kant had given it. Mathematics deals with quantity, with magnitudes and their external relations. Its method is to hold its objects fixed and move through chains of identical substitutions. A proof in geometry proceeds by replacing one expression with another that is equal to it, step by step, until the desired conclusion appears. This method works because its objects are inert, indifferent to one another, and devoid of inner development. Numbers do not contradict themselves. Figures do not develop. And it is for this reason, Hegel contends, that mathematics is inadequate for philosophy.

Philosophy requires the concept (Begriff), which is self-moving, self-differentiating, and self-returning. The concept does not remain fixed while the thinker manipulates it from outside; it develops through its own internal contradictions. Mathematical thinking cannot grasp this movement because it operates by the understanding (Verstand), which fixes distinctions and holds them apart. Reason (Vernunft), by contrast, perceives how opposites pass into one another and are reconciled at a higher level. To attempt philosophy in the mathematical manner, as Spinoza did with his geometrical method, is, in Hegel's view, to impose an alien form on content that has its own life. The distinction between the understanding and reason bears upon similar questions treated under the idea of Dialectic.

Hegel's critique extends to the mathematical concept of infinity. The mathematical infinite, he argues, is the "spurious" infinite: an endless progression, one number after another, never complete. The true infinite is not a line stretching without end; it is the self-enclosed totality that contains its own negation and returns to itself. Mathematics can approach this notion, as in the calculus with its vanishing quantities, but cannot think it adequately, because mathematical form keeps quantity external and fixed. The philosophical treatment of the infinite is considered further under the idea of Infinity.

Hegel's limitation of mathematics to the domain of external, quantitative relations raised a question that nineteenth-century mathematics would press in unexpected ways. Cantor's theory of completed infinite sets seemed to require the kind of actual, qualitative infinity that Hegel had claimed mathematics could not think, leaving open the question whether Hegel was mistaken about the scope of mathematics or whether the new mathematics had outrun its own foundations.