Chapter 62
Mathematics: number, geometry and calculation; mathematical method and certainty.
143 passages · 37 principal · Covers the 49 volumes of primary texts
Mathematics, in the sense these books give it, takes in number and figure, the arts of counting, measuring and calculating, and the claim of mathematical reasoning to a certainty no other knowledge enjoys. Hobbes classes geometry and arithmetic among the sciences that draw consequences from quantity and determined motion1. Augustine, when he wants a standard of what it would be to know anything clearly, names the sum of seven and three2. Hume holds that the truths Euclid demonstrated would remain certain even if no circle or triangle existed in nature3. Mathematics is a great idea for these books because nearly every author who asks what knowledge is, how far it reaches, and how it should be gained measures his answer against it. Some treat it as the pattern of all demonstration, others as a warning about what demonstration cannot do.
The ancients speak of mathematics as a gift or a discovery. Aeschylus’s Prometheus counts Number, “chief device of all,” among the arts he gave to mortals along with letters4. Herodotus offers a more worldly origin: geometry arose in Egypt from the need to remeasure land after the river’s floods and passed from there to Greece5. Later writers keep something of both views. Newton grounds geometry in the mechanical drawing of lines and circles6. Hobbes notes that numbering itself depends on numeral words7. Huxley observes that the Greek foundations of geometry still teach our children two thousand years later8.
For Plato, number matters less as a science in its own right than as a way into the nature of ideas. In the Phaedo, Socrates is puzzled that one added to one should become two by mere juxtaposition, and that dividing one should also produce two9. He then uses odd and even to show how a thing that shares in one idea must exclude its opposite: three can never become even10,11. Earlier in the dialogue, a diagram shows that someone who is merely questioned can produce right answers from within himself, which is offered as evidence that knowledge is recollected12. Augustine carries the inwardness further. The numbers and dimensions in his memory, he says, are not images of any lines he has seen or things he has counted13. Pascal grants that numerical proportions are eternal truths depending on a first truth called God, but he denies that knowing them does anything for salvation14. The moderns contest the status of these objects. Berkeley allows that mathematicians consider quantity without regard to sensible qualities, but he denies that this proves there are pure abstract ideas15. Mill, against the intuitionists, sets out to explain even the necessary truths of mathematics from experience, attacking what he takes to be their stronghold16. Whether mathematical objects are recollected forms, inner reasons, abstractions, or generalizations from experience remains an open question in these pages.
The authors agree far more on the fact of mathematical certainty than on its ground. Simmias already remarks that arguments from mere probability would deceive in geometry17. Hobbes calls arithmetic a certain and infallible art, though those who practise it may err18, and he illustrates the point with the triangle whose angles equal two right angles19. Descartes traces the certainty of geometrical proofs to clear conception. He adds that such proofs do not by themselves assure us that their objects exist20, yet he still appeals to the force of mathematical demonstration to separate true reasons from mere likelihoods21. Hume agrees that these sciences are independent of existence. Their ideas are clear and determinate, and their difficulty lies only in long chains of inference22. Mill finds that the theorems of geometry look different from the definitions and axioms but are all contained in them23. Kant, by contrast, uses the bisection of a line by intersecting arcs to show that mathematical propositions about means are synthetic24. Mill draws a practical consequence from this certainty. Because in mathematics all the argument lies on one side, teaching suffices, and objections need not be rehearsed as they must be in disputed subjects25. Pascal puts it most simply: mathematics has a known object, which is proofs26.
It is Pascal who gives the fullest account of geometrical method. Geometry alone, he argues, has firm and agreed principles, because it deals with simple figures27. It alone teaches true demonstration, and its simple precepts suffice where other rules are useless or even harmful28,29. Those who truly know the method are few in any nation30, and he calls his own rules the geometrical proof of the art of persuading31,32. The heart of his account is a paradox. A perfect order would define every term and prove every proposition, and this is impossible. Geometry therefore follows a middle order, assuming only what is clear by natural light. That order is less convincing than the perfect one but no less certain33. It leaves space, time, motion, number and equality undefined34,35. These terms cannot be defined because they are extremely obvious, and their lack of definition and proof is a perfection, not a defect36,37. Definitions of names, such as calling numbers divisible by two even, are free stipulations38. Pascal does not hide that the foundations remain bare: terms like centre and motion are never established by proof39.
Whether the method can be carried beyond quantity divides these authors more sharply than anything else. Descartes applies it to problems in mathematics and to other questions he makes almost mathematical by detaching them from uncertain principles40. Kant draws an analogy between pure and applied mathematics and pure and applied morals41. Hume answers that quantity and number are the only proper objects of demonstration, and that extending it further is sophistry42. Hobbes gives a political explanation of why geometry is undisputed: its truths cross no man’s ambition or profit, and if they did, the books of geometry would be burned43. Mill rejects pure geometry as a model for politics because it is not a science of causation44. Freeman contrasts the mathematician’s absolute demonstration with the uncertainty of historical fact45. Burke notes that a common measure gives mathematics a certainty denied to matters of degree such as smoothness or shades of colour46.
Pascal turns the same contrast into an account of two kinds of mind. Mathematical principles are palpable but unfamiliar to ordinary habit47. Mathematicians are lost where principles cannot be set in order, and intuitive minds cannot bend themselves to mathematical principles48,49. Pure mathematicians are exact only when definitions and axioms make things clear50. Their art requires holding many premises at once51, and it belongs to intellect rather than to intuition52. Others see a discipline that can pass beyond the subject itself. Voltaire says Locke had little skill in calculation yet had a geometrical head without geometry53. Bacon prescribes mathematics for a wandering wit, because any lapse of attention forces one to begin again54. Franklin, less admiringly, describes a mathematician who demanded universal precision in ordinary speech55.
The infinite is where mathematics both shows its strength and raises doubt. Pascal argues that geometry rests on the divisibility of space without end. A supposed indivisible would have no extent at all56, and an argument about squares and points shows that space is not composed of finite indivisibles57. Units compose numbers because unity is of the same kind as number: multiplied, it exceeds any number. An indivisible, however multiplied, never makes an extension, just as zero never makes a number, an instant never makes time, and rest never makes motion58,59,60. Those who deny infinite divisibility, he concludes, cannot claim geometrical demonstrations at all61. For Pascal the indivisible point is only a limit of the senses62. He also draws on infinite number to speak of God and man. Unity added to infinity changes nothing63. Infinite number is neither even nor odd, though every finite number is one or the other64. Intuition knows that number is infinite before reason proves anything about square numbers65. Berkeley and Hume take the same paradoxes in other directions. Berkeley holds that denying absolutely extended things clears away the puzzle of how a finite extension can be infinitely divisible66, and that difficulties about incommensurables and asymptotes do not undo mathematical demonstration67. Hume finds that demonstrations about circles and tangents appear beyond objection yet lead to contradiction68, so that geometry, the most certain of sciences, becomes a ground for scepticism69.
The story of mathematics applied to nature runs from complaint to triumph. Copernicus objects that the astronomers’ calculations cannot fix even the length of the year and lack consistent principles70. Newton applies mathematics to natural philosophy in order to demonstrate forces and motions6. Locke praises the Principia for giving demonstrated knowledge of particular provinces of nature71. Voltaire shows Newton reaching great discoveries with a quadrant and a little arithmetic, computing the moon’s orbit and fall72, the quantity of matter in the sun and planets73, and the properties of light74,75. Hume adds a caution: geometry helps natural philosophy apply its laws but cannot discover those laws or their ultimate causes76. Nineteenth-century science bears out both the power and the limit. Faraday explains the inverse-square relation through a proportion of squares77. Kelvin presents the mathematical details of the wave theory of light78 and derives wave-lengths by proportion from measured distances79. He credits Laplace with the true dynamical principles of the tides80,81, and he analyses the tide into simple harmonic constituents whose sum a machine can calculate82,83,84. Helmholtz computes the heat of glacial friction85, and Newcomb uses cubes and cube roots to estimate the bounds of the stellar universe86. Emerson marvels that geometry, a pure abstraction of mind, should measure the motions of the planets87.
The method itself also changed. Descartes found geometrical analysis tied to figures and tiring to the imagination, and algebra confusingly bound to rules and formulas88. He therefore studied proportions in general form, represented by lines and brief characters89, and joined geometry to algebra by following the order of his method90. Voltaire tells how Descartes expressed curves by equations91, and how infinite series and the quadrature of the hyperbola led to Newton’s calculus and its priority disputes with Leibniz92,93.
Calculation also enters practical life and nature. Harvey multiplies drachms by pulses to demonstrate that the blood circulates94. Pascal describes his machine that performs arithmetic without pen or counters95. Franklin estimates the size of a crowd by imagining a semicircle96. Dana’s shipmate Harris delights in calculating hides, canvas and dead reckoning, and outstrips the narrator, who had studied mathematics97,98. Hume quantifies probability with dice99, Smith shows that buying more lottery tickets brings loss closer to certainty100, and Geikie estimates rates of erosion by simple proportion101. Darwin finds that bees solve the geometrical problem of storing the most honey with the least wax102,103. Voltaire marks a limit to usefulness: elementary arithmetic brings wealth in trade, while the algebraist’s refinements merely delight104.
Society and taste resist this model. Smith reports Quesnay’s system cast in arithmetical tables105. Burke mocks the French for dividing their country into squares, a task that needs only a surveyor106. Equal squares, he observes, yield unequal power107, and even an exact geometry would ignore moral considerations108. Mathematical speculation leaves the imagination unmoved, so geometry cannot explain beauty109.
Education reflects the value placed on mathematics. Augustine mastered geometry and arithmetic without a teacher110, while Harrison laments that the quadrivial arts are now little regarded111. Milton would teach arithmetic early, even in play, and lead it on to fortification and navigation112,113. Locke calls arithmetic the easiest abstract reasoning and of universal use114. He would begin it once geography is fixed in memory115 and thinks six books of Euclid suffice for a man of business116. Mill learned Euclid, algebra and some calculus largely alone117,118. Opinions of its worth vary. Copernicus says mathematics is written for mathematicians119, and Hobbes calls geometry the only science God has bestowed on mankind7. The young Descartes admired its certitude but thought its use merely mechanical120. Pascal thinks geometry a natural greatness that deserves esteem but confers no social precedence121,122, and notes that it is studied by more people than the study of man123.
Literature, finally, makes number into image and jest. Dante places Euclid among the great124 and likens himself to a geometer vainly trying to square the circle125. Jonson measures quarrels by angles126, and Burns proves immortality from Euclid in jest127. Goethe’s witch recites a nonsensical multiplication table128. Browne reveres the magic of numbers129, while Darwin traces numerical parallelisms in classification to arbitrary valuation of groups130. Emerson likens compensation to an algebraic equation131. Some questions remain unsettled across the whole conversation: whether mathematics is recollected or learned from experience, whether its certainty can guide morals and politics, and whether the infinite confirms reason or undermines it.
Introductory essay written by Claude Opus 5.5 from the outline and the notes on every passage below; quotations are checked against this edition.
See also: Reasoning 2b · Logic 3c · Principle 1a
See also: Mind 2b
See also: Astronomy and Cosmology 4b · Mechanics 3b
See also: Education 8f
See also: Reasoning, Logic, Definition, Infinity, Knowledge, Principle