Charles Davenport (1866-1944) was a distinguished American biologist. He was also an infamous champion of eugenics, on whose preposterous basis he promoted the segregation and sterilization of “inferior” individuals and lobbied in favor of a U. S. federal law preventing immigration from Asia and setting rigid immigration quotas from Eastern and Southern Europe. (Signed by Calvin Coolidge in 1924, this law was revised in 1952 and finally rescinded in 1965 by Lyndon Johnson.) While arrogantly carrying on in public, he proffered: “People don’t have heated discussions on the multiplication tables. They will not dispute quantitative findings in any science.”1placeholder Here I will say nothing more about him; his only relevance is the sentence I quoted, and pointedly its mention of multiplication tables, which belong to mathematics. Like many others before and after him, Davenport was using something of the sort to claim “mathematical” certainty for his theses. And it is this certainty that I intend to call in question and reinterpret here.2placeholder
To begin with, I distinguish two aspects of mathematics. First, it is supposed to arrive at necessary, indisputable, definitive conclusions—at final, permanent resting points—; and such an image of it has regularly been used to cow those other disciplines (including philosophy) which proceed tentatively and are often forced to backtrack.3placeholder The reason for this special character of mathematics is that it provides knowledge a priori—prior to, and independent of, any experience—and hence it (and the knowledge it provides) is not subject to the vagaries of the empirical world. Kant says that “universal cognitions, which at the same time have the character of inner necessity, must be clear and certain of themselves, independently of experience; hence one calls them a priori cognitions,”4placeholder and that “[m]athematics gives us a splendid example of how far we can go with a priori cognition independently of experience.”5placeholder In the entry “Philosophy of Mathematics” of the Stanford Encyclopedia of Philosophy,6placeholder Leon Horsten generalizes on Kant’s position by writing: “The received view has it that mathematical proofs yield a priori knowledge,”7placeholder and only considers objections to this view based on the use of computers (to which I will return) in generating proofs.8placeholder
To get to the second aspect of mathematics I want to highlight here, look at chess: an example and a term of comparison that recurs in what follows. Say that, after carefully analyzing the situation on a chessboard, I conclude that White is in an irremediably losing position. If my analysis is correct, there is no way anyone can prove me wrong: not now, not ever. What I have concluded has the universality and necessity Kant claimed for mathematics, and it has them for the same reason mathematics has them: because it is known a priori, independently of any experience. But chess is supposed to be a game played for its own sake: what is true of it is so only within its artificially set boundaries (though see below). Mathematics, on the contrary, is supposed to inform us about reality: not just the reality (whatever that might be) of numbers and geometrical figures but reality, period—the reality of natural and social environments. And it is in this sense and for this reason that Davenport appealed to its authority. If we run a mathematical proof on a blackboard, we expect its conclusion to apply well beyond that board and the signs we traced on it, or even the “game” where the signs belong, and to extend wherever it applies its universality and necessity. Which, besides conferring presumed prestige on theses like Davenport’s, and many other less infamous ones, is also mysterious: how can a science that has nothing empirical about it, that does not call on any data or observations, be to such an extent depictive and predictive of the empirical world?
If the certainty of mathematics has two aspects, so does my attempt at recontextualizing it. On the one hand, mathematics is indeed a game, as chess is, and is useful and informative in the way games can be. Within it, as within any other game, we construct and manipulate models of reality, and can apply what we learn from them to the reality they model. We don’t regard it as a mystery if the toy train we played with for a while teaches us something about how to operate a real train, or if gambits explored on the chessboard prove instructive in concocting maneuvers at the office. But, in order to approve this deflationary explanation of mathematics’ “success,” we need to undermine its pretense of extending universality and necessity beyond its immediate domain. We need to appreciate the fact that, as there is always going to be a gap between what the toy train teaches us and the workings of a real train, or chess gambits and a careerist’s intrigues,9placeholder there is always going to be a gap between what a mathematical proof teaches us and the details of any real environment. A smaller gap in the latter case than in the former; but that is only a reflection of how much longer and more ingeniously we played one game than the others. And in any case the difference between the two gaps is an empirical matter: it does not change the nature and quality of what is at stake. Nor does it change the fact that there is no mystery in the mathematical game providing a (tentative) road map for the empirical world.
One last remark is in order before we proceed. The mathematical examples I will discuss are simple ones (not quite as simple as multiplication tables, though not much harder either), which may make my argument snubbed by experts. But the sorts of complications that would please the experts would hide the essential structure on which they work, as getting to the deepest implications of a knight opening in chess would hide the general function a knight can have in mimicking (imperfectly, to be sure) real objects and events. I am not a professional mathematician, and will make no effort to look like one; but what I am going to say does not require this status, nor does understanding it.
1. Proofs
We read in Diogenes Laertius (on testimony from Apollodorus) that Pythagoras, when he proved the theorem that goes by his name, offered as a sign of his gratitude to the gods a hecatomb: a sacrifice of a hundred oxen.10placeholder It is dubious information, since Pythagoras was a vegetarian and a believer in metempsychosis (the doctrine of the transmigration of souls), so for all he knew some relative of his could have been incarnate in one of those oxen. But the scale of the offering is not surprising: Pythagoras’ is the first mathematical proof of which we have notice, and that such a proof was even possible might have seemed a miracle, for which it was appropriate to pay massive homage to the dwellers of Mount Olympus.
There is a traditional narrative on the origins of mathematics in the West. For Babylonians, it says, mathematics consisted in the formulation and resolution of concrete problems: it was a collection of calculating procedures, empirical calculating procedures. Then came the Greeks (“that admirable people”) and things went through a radical change: mathematics became a purely rational science, in which consequences followed from premises, and conclusions were established, with argumentative necessity. From the a posteriori condition of an ensemble of empirical procedures it moved to the supernatural arena of the a priori. Pythagoras opened the way and Plato traveled it, being inspired by the lofty status of mathematics to build that vision of his of which all subsequent European philosophy (according to Alfred North Whitehead, a mathematician and a philosopher) is but a series of footnotes.11placeholder In the last few decades, we may be witnessing (as Horsten notes) a reversal of this course: by using more and more powerful computers, we are able to attack problems for which we had no provable (a priori) solution, and to find for them some other “solutions” that, if they are not exact, are at least practicable. We seem to have returned to something similar to Babylonian mathematics, and to have exited the safe ground of the a priori. I will contest this narrative here; but, before I get to my criticism, I spell out more of the narrative—of what it suggests took place in the long interlude between Pythagoras and the late twentieth century.
The Pythagorean theorem states that, in a right triangle, the square built on the hypotenuse has an equal area to the sum of the squares built on the other two sides. We have numerous alternative proofs of it; whichever we choose, the conclusion is inescapable, and nothing more need be said on the matter (at least within the geometry where Pythagoras was probably working, eventually called Euclidean). Deeply influenced by such proofs, Plato recommended ten years of mathematical study for the future governors of his ideal republic, before having them turn to his dialectic, which shared absolute necessity with mathematics but was even better in not requiring, as mathematics does, arbitrary axioms from which to get started in the demonstrative process (such as those, we would say today, that distinguish Euclidean geometry from non-Euclidean ones). He was, however, quite negative on the prospect that the necessary, eternal truths of mathematics (or dialectic) could have any direct application to the spatiotemporal world in which we live our everyday lives. That was for him a world of shadows projected on the wall of a cave, of ghosts constantly appearing, disappearing, and mutating, where the precision of mathematics (and dialectic) had no currency. The best one could hope for was that mathematics (and dialectic) be used as a guide within spacetime: that we could recognize in it imperfect copies of the ideal models provided by mathematics (and dialectic), and be able to approximate their precision and issue value judgments on what approximations were more faithful than what others. The ideal republic is not to be found in the empirical world; but using it as a standard we could judge Sparta, say, to be a better State than Athens. The perfection of (Euclidean) triangles or squares is also not to be found there; but using those perfect figures as models we could get a better organization of cultivated areas in a field, say, than by using tradition and guesswork.
Over two thousand years later there appeared to be little evolution on this topic. Here is David Hume, in An Enquiry concerning Human Understanding (1748):
“All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation, which is either intuitively or demonstratively certain. That the square of the hypothenuse [sic] is equal to the square of the two sides, is a proposition, which expresses a relation between these figures. That three times five is equal to the half of thirty, expresses a relation between these numbers. Propositions of this kind are discoverable by the mere operation of thought, without dependence on what is any where existent in the universe. Though there never were a circle or triangle in nature, the truths, demonstrated by Euclid, would for ever retain their certainty and evidence.
Matters of fact, which are the second objects of human reason, are not ascertained in the same manner; nor is our evidence of their truth, however great, of a like nature with the foregoing. The contrary of every matter of fact is still possible; because it can never imply a contradiction, and is conceived by the mind with the same facility and distinctness, as if ever so conformable to reality. That the sun will not rise to-morrow is no less intelligible a proposition, and implies no more contradiction, than the affirmation, that it will rise. We should in vain, therefore, attempt to demonstrate its falsehood. Were it demonstratively false, it would imply a contradiction, and could never be distinctly conceived by the mind.”12placeholder
The contrast set by Hume between relations of ideas and matters of fact matches the one set by Plato between the mathematical and the spatiotemporal worlds. On the one hand, for both, are eternal and unassailable truths, to be established by reasoning a priori; on the other, empirical turbulence we cannot control with certainty but can only manage with the help of perhaps reassuring but also vacillating habit (prisoners in the Platonic cave, too, developed habits in predicting the moves of the shadows projected on the wall, and went by them in living and occasionally tyrannizing those who would want to teach them any better). The two worlds (or two sets of “matters”) are heterogeneous and do not communicate; mathematics may inspire us (as it did Plato), but its standards (as in Plato) are not met within everyday life (there might never be “a circle or triangle in nature”).
Hume, however, was no mathematician, so he was not aware that by his time people in the field had formed vaster ambitions and were prepared for stronger statements. Crucial in this regard—indeed of monumental importance for my purposes here—is Galileo, regarded by both Albert Einstein13placeholder and Stephen Hawking14placeholder as the father of the “new science” that arose in the seventeenth century and within whose scope we are still operating today. In his Dialogue concerning the Two Chief World Systems,15placeholder he agrees with Plato in asserting the necessary truth of mathematics—or at least, as far as humans go, of as much of it as they are capable of comprehending:
“[T]he human understanding can be taken in two modes, the intensive or the extensive. Extensively, that is, with regard to the multitude of intelligibles, which are infinite, the human understanding is as nothing even if it understands a thousand propositions; for a thousand in relation to infinity is zero. But taking man’s understanding intensively, in so far as this term denotes understanding some proposition perfectly, I say that the human intellect does understand some of them perfectly, and thus in these it has as much absolute certainty as Nature itself has. Of such are the mathematical sciences alone; that is, geometry and arithmetic, in which the Divine intellect indeed knows infinitely more propositions, since it knows all. But with regard to those few which the human intellect does understand, I believe that its knowledge equals the Divine in objective certainty, for here it succeeds in understanding necessity, beyond which there can be no greater sureness.” (103)16placeholder
Then, however, he makes a decisive step beyond Plato: mathematics is not to speak only of perfect shapes that are nowhere to be found in the empirical world and can only be approximated there (of Euclidean triangles and squares, say, or of the five regular solids17placeholder), but of all shapes, however irregular they might be, and is to study them and get to know them all equally. Whereas the previous passage is spoken by Salviati, who is Galileo’s mouthpiece, this other revolutionary claim is wriggled by the more neutral Sagredo out of the polemical objective Simplicio:
“Sagredo: [S]ince the irregularity is the cause of the difficulty, even the shape of a rock broken at random with a hammer would be among the shapes hard to produce, this being perhaps more irregular than a horse [an example he himself had brought up]?
Simplicio: It should be as you say.
Sagredo: But tell me: Whatever form this rock has, does it have this perfectly, or not?
Simplicio: That which it has, it has so perfectly that nothing else corresponds to it so exactly.” (209-210)
This exchange, however, issues within the lengthy discussion of a remark Salviati had made a few pages earlier: a perfect sphere will touch a perfect plane in exactly one point, and the spheres and planes of ordinary life touch more widely than that; don’t worry, though, because “abstract” mathematics has as much room, and can provide as much understanding, for imperfect spheres and planes as for perfect ones.
“I tell you that even in the abstract, an immaterial sphere which is not a perfect sphere can touch an immaterial plane which is not perfectly flat in not one point, but over a part of its surface, so that what happens in the concrete up to this point happens the same way in the abstract. It would be novel indeed if computations and ratios made in abstract numbers should not thereafter correspond to concrete gold and silver coins and merchandise. Do you know what does happen, Simplicio? Just as the computer [that is, the accountant] who wants his calculations to deal with sugar, silk, and wool must discount the boxes, bales, and other packings, so the mathematical scientist (filosofo geometra), when he wants to recognize in the concrete the effects which he has proved in the abstract, must deduct the material hindrances, and if he is able to do so, I assure you that things are in no less agreement than arithmetical computations.” (207)
By combining the two claims thus made, we are told that mathematics (here, geometry—on whose primacy at the time I will comment below) can directly address all empirical shapes with the same absolute certainty enjoyed by the Pythagorean theorem: a certainty of which not even God could do better. The calculus, to be discovered shortly thereafter by Newton and Leibniz, was to (apparently) vindicate this forceful commitment, and as such I will take it up later. But first let us go back to the Greeks.
2. Trigonometry
Aristotle thought that the universe was divided into two, radically distinct, parts. Within the Earth’s sphere, which occupied its center, all that existed was constituted of the four elements earth, water, air, and fire, and was subject to change: to generations, corruptions, and alterations. Within all other spheres surrounding (and revolving around) the Earth, where all heavenly bodies (including the Sun) were located, everything was constituted of a single, unchangeable element: aether. Those heavenly bodies, then, were themselves unchangeable, and had unchangeable motions: whereas on Earth bodies could go through intermittent, violent motions, that forced them away from their natural locations and then brought them back there by the shortest, straight-line route, after which they could rest indefinitely (until violence was again exercised on them), in the heavens bodies had natural motions—circular, eternal, and eternally self-identical. It makes sense, therefore, that in the heavens mathematics should have its most reliable application: to beings which, though material, were otherwise the most similar to Platonic ideas. And that in studying the heavens should be developed, more by astronomers than by pure mathematicians, the first mathematical discipline that directly applied to reality: trigonometry.
Hipparchus of Nicaea, who lived in the second century B. C., was the founder of this discipline and (for many) the greatest astronomer of antiquity (or, for others, the very father of astronomy). Though virtually nothing is left of his works, he is thought to have provided the first accurate models of the Sun’s and the Moon’s motions, the first precise estimate of the distance between the Earth and the Moon, the discovery of the precession of the equinoxes, the first comprehensive star catalogue of the Western world, and possibly the invention of the astrolabe and of the armillary sphere. What we know of him comes from Strabo’s Geography and Pliny the Elder’s Natural History, as well as, most significantly, from the Almagest of his colleague Claudius Ptolemy (an Egyptian astronomer and mathematician of the second century A. D. who wrote in Koine Greek): the most systematic ancient account of the geocentric model of the solar system—one of the two “chief world systems” treated in Galileo’s dialogue. Let us now get to what motivated his founding of a discipline.
Imagine you are observing the motion of a heavenly body from Earth. You believe that motion to be circular; but what you can actually observe is successive locations of the body, and the segment of a straight line that connects them as the body moves from one to the other, not a circular arc. Introducing the relevant technical term, you are observing the chord of the arc traveled (as you believe) by the heavenly body, where a chord is any line segment joining two points of a circumference, and subtending an arc, as in this figure:
A fundamental problem of ancient astronomy was that of establishing a correlation between the lengths of arcs and of the chords that subtended them—or, equivalently, since every arc corresponds to a central angle, between the lengths of the chords and the amplitudes of the corresponding central angles. Hipparchus was the first to do this, but Ptolemy was the first to deliver, in the Almagest, a table of chords, which gave their lengths for half-degree increments of the amplitudes of the corresponding central angles. Starting with Indian astronomers of the fifth century, chords were replaced by sines (a sine is half a chord) and tables of sines were also offered. Which seemed to make astonishing achievements possible. For an illustration of which, we briefly migrate to the American West.
On the morning of October 18, 1805, the expedition led by Meriwether Lewis and William Clark, the first one conducted from the U. S. and able to reach the Pacific Ocean by land, was at the confluence between the Snake and the Columbia rivers, in today’s Washington State. Neither river was crossed by a bridge, and Clark set himself to measure the width of both. With reference to the following figure, he chose a point A at the confluence and two other points B and D on the Columbia and on the Snake, respectively, and measured the distances between A and B, and between A and D. Then he chose two conspicuous landmarks (large rocks or trees) C and E on the other sides of both rivers and, with a circumferentor (a six-inch-diameter surveying compass), measured the angles between AB and AC, AB and BC, AD and AE, AD and DE. For the Columbia, the angles turned out to be, respectively, 22° and 28.5° (from now on, I will disregard the Snake, which would add nothing new). Though Clark does not tell us how he went from there, he had enough to attain his goal. For, knowing three elements of the triangle ABC (two angles and one side), he could calculate all the others. How? By using trigonometry.
Here is a way he could have proceeded: With reference now to the following figure, the green line forming the angle α cuts the circumference at P, and the value of the ordinate for P (in red in the figure) is the sine of α (the abscissa, in blue in the figure, is its cosine).
The law of sines states that, in every triangle, there is a constant ratio between the length of a side and the sine of the opposite angle; therefore, if we know the amplitudes of two angles (and, by implication, of the third one as well) and the length of one side, we can calculate the lengths of the other two sides. And, if we know all sides, we can calculate the area of the triangle by using Heron’s formula (which follows from the Pythagorean theorem): if p is the semi-perimeter of the triangle and a, b, c are the three sides, then the area is the square root of
p(p – a)(p – b)(p – c).
Given the area and the side AB (on Clark’s side of the river), we can use that side as the base of the triangle and calculate its corresponding height, which gives us the width of the river. All of it without getting our feet wet! Magic? No wonder mathematics always elicited so much respect! But let us examine this case more carefully.
Clark’s instruments were somewhat rough, hence, when it came to what was on the other side of the river, he had to use conspicuous landmarks that were only close to that side but not on it. It was good enough for his purposes; and his practice was not substantially different from what a contemporary engineer would do when building a bridge on the river. Though having much more sophisticated tools available, he would be content if the bridge satisfied required specifications to a high extent, and would not be bothered if the outcome still only approximated those specifications. All of which is consistent with Plato’s picture: we will never find the exactness of mathematics in our spatiotemporal world; we will only ever find imperfect copies of it. But suppose we go beyond that picture, to settle on Galileo’s: every feature of the spatiotemporal world can be studied with mathematical exactness; maddeningly complicated features of it will be captured by maddeningly complicated mathematical structures. And never mind what engineers do: the tools we have available today may afford a lot of credibility to the Galileian picture. By using them, we can determine the points we need for a Clark-kind of triangulation with perfect accuracy, and the result we obtain by calculating on their basis will be similarly (perfectly) accurate. We may thus have eliminated the approximation by which mathematics applies to the spatiotemporal world. Only, however, to bring to the fore another ineliminable approximation, this one internal to mathematics itself.
The function sine is not defined for most angles, and it is not for the angles measured by Clark. It is for a small number of notable angles: the sine of 0° is 0; the sine of 30° is 1/2; the sine of 90° is 1. The sine of 28.5°? One way of calculating it was discovered in 1715 by the English mathematician Brook Taylor (though it had been found half a century earlier, and ignored until then, by the Scottish mathematician and astronomer James Gregory). It requires consideration of the following operation (a Taylor series expansion):
sin(x) = x – x3/3! + x5/5! – x7/7! + …,
where n! (factorial n) = 1 × 2 × 3 × … × n and the value of x is in radians, which I introduce in the next section. The ideal completion of the operation would be a irrational number, which also comes up in the next section; here I point out is that the operation never ends,18placeholder so all we ever get is an approximation to the required value (with eight decimal places, it would be 0.47715876). And I repeat: this approximation is internal to mathematics, not a consequence of its application to the unstable world of rivers and mountains. Mathematics cannot say what the sine of 28.5° is; and, instead of determining that value, it proposes a calculating procedure, which, the longer you pursue it, the closer it gets you to a desired destination which is, however, unreachable. It looks as if the decisive step from Babylonian calculations to Greek demonstrations was never actually (fully) made, and no reverse trend is only now under way, in the age of computers (in fact, a Taylor series is what most computers use to calculate the sine function, in total continuity with human operators). It looks as if, with the exception of a few notable mathematical objects that attracted more than their due share of attention and on which a whole ideology was built, what mathematics (not the application of mathematics to a non-mathematical world) was all about, no less with the admirable Greeks than with their predecessors, was calculations that never reached a conclusion—for which one could never write “the pompous abbreviation ‘QED’.”19placeholder It was calculations that had none of the necessity pertaining to a priori knowledge: every step in them issued from an operation whose result was only known after it was (empirically) performed,20placeholder and was in any case only partially right (hence, in fact, wrong) with respect to the desired mathematical goal.
3. Irrationality and transcendence
My assertion that the sine of 28.5° is not defined will probably be questioned, and so will the sharp contrast I drew between providing that definition and providing an interminable calculating procedure; so I need to get deeper into this matter, and I will begin to do so by bringing up the Greeks once more.
The Pythagorean theorem lets us easily calculate (or so it seems) the diagonal of a square. For, with reference to the following figure,
it is easy to verify that ABC and ACD are right equilateral triangles. Therefore,
AB2 + BC2 = AC2,
that is (since AB = BC),
2AB2 = AC2
Thus the diagonal we are looking for is √2AB, and, assuming that AB has length 1, is √2. Problem solved? Not quite, because what is √2?
The ancient Greeks only admitted as numbers those we call today rational ones, where the Latin ratio is understood as quotient: fractions with natural numbers as both numerator and denominator. And we can prove that there is no such fraction equal to √2.21placeholder For suppose there is (the proof is by reductio ad absurdum), imagine it reduced, and name it p/q. Then
p2/q2 = 2.
Since p/q is reduced, it is impossible that both p and q be even; so either they are both odd or one of them is even and the other is odd. If p is odd then p2 is also odd, and it is impossible that the result of dividing an odd number by any other number (even or odd) be 2. If p is even then q is odd and, for some r, p is 2r; then p2 is 4r2, and it is impossible that the result of dividing 4r2 by an odd number be 2.
The above means that, for the Greeks, there existed no number equal to √2. But in geometry the diagonal of a square and its relation to the square’s side can be seen, and what we see can be used to prove important properties of this diagonal. For example, with reference to the following figure, we can prove that in a square the two diagonals divide each other exactly in the middle and are bisectors of the relevant angles:
This discrepancy was taken, for a long time, to be evidence of the superiority of geometry over arithmetic. Says Leonardo in the second Madrid codex: “the deaf [that is, irrational] roots are incommensurable and non-communicating. And for this reason manifestly the geometrical proportion is of greater abstraction and consideration than the arithmetical one. For see that it considers both the rational and the irrational, as was said, whereas the arithmetical one [considers] only the rational, which can be named by some number. Of the other one it is totally deprived.”22placeholder Thus, while arithmetic hobbled, geometry was regarded as the archetype of logical and scientific rigor; well into the seventeenth century, Spinoza was to entitle his masterpiece Ethica ordine geometrico demonstrata, and to order it, indeed, in strict analogy with Euclid’s Elements—with definitions, axioms, postulates, and propositions. A few decades after Spinoza, the calculus seemed to revolutionize these priorities, and I said that I will get to it later. For the moment, I return to the issue we are discussing now.
Calculating a square root is an operation that, in most cases (all cases except those numbers, like 1, 4, 9, 16, 25, … that are perfect squares), does not give a definite result. Let us try to understand it, starting from geometry and considering the number (not a perfect square) 97. Geometrically, to get a square root is to get the side of a square of which we know the area. In the following figure
the whole square represents the square with side 10 (the smallest natural number whose square 100 is greater than 97) and the green square represents the square with side 9 (the greatest natural number whose square 81 is smaller than 97). The square with area 97 includes the green square, the two blue rectangles of which we know the basis (9) but not the height, and the red square whose side is equal to the (to us unknown) height of the rectangles.
The side we are looking for must measure between 9 and 10, so it must be 9 followed by a decimal expansion. Let us begin by calculating the first decimal digit. The difference 16 between 97 and 81, which we take to be expressed in squared natural numbers, becomes 1600 when expressed in squared tenths of natural numbers. Let us now imagine to line up the two rectangles and the red square, that together have area 1600. Of the resulting rectangle we don’t know the height and we know that the basis must be at least 180 (90 + 90 + x; the original basis 9 of the rectangles becomes 90 when expressed in tenths of natural numbers); so let us try various heights. If the height was 9 then the basis would be 189 (90 + 90 + 9); but 189 × 9 = 1701, which is larger than 1600. On the other hand, 188 × 8 = 1504; therefore, the first decimal digit must be 8. Then we must repeat the same procedure for the new rectangles and little square generated from the difference between 97 and the square 96.04 of 9.8, and we will obtain the second decimal digit 4. And on and on. The procedure will never stop. √97 will be forever (better) approximated but never reached.
If we apply this procedure to 2, we obtain in succession the following partial results:
1
1,4
1,41
1,414
1,4142
1,41421
.
.
.
And, imagining to have completed this infinite calculation, the final result would be the irrational number √2.
The notion of an infinite procedure that gets completed is an irksome one; so we will not be shocked to find out that, though irrational numbers had been a constant presence in mathematics for at least two centuries, one had to wait until 1872 to be told what an irrational number is. And then one was told with a vengeance: in that year four different authors (Georg Cantor, Richard Dedekind, Edward Heine, and Karl Weierstrass; Charles Meray had taken something similar to the Heine approach in 1869) offered definitions of it. Here I present Dedekind’s, from his Stetigkeit und irrationale Zahlen.23placeholder
A Dedekind cut is an ordered pair of non-empty, disjoint sets A and B such that (a) all rational numbers belong to either A or B, (b), for all elements a of A and b of B, a < b, and (c) A has no greatest element. If B has a smallest element, the cut is a rational number; otherwise, it is an irrational number. For illustration, A might contain all numbers smaller than 1/2 and B contain a smallest element 1/2; then the cut coincides with the rational number 1/2. Or A might contain all rational numbers smaller than √2 and B all rational numbers greater than √2. The previous description of calculating √2 shows that A contains
1
1,4
1,41
1,414
1,4142
1,41421
.
.
.
and B contains
2
1,5
1,42
1,415
1,4143
1,41422
.
.
.
The relevant cut, that is, the ordered pair <A, B> of these two infinite sets, is the irrational number √2. A calculation never reaching an end (and therefore being potentially infinite—as, say, the series of prime numbers is: in the sense that, for every prime number, we can construct a larger one, without end24placeholder) is thus replaced by a pair of actual infinities: infinities which are given once and for all, which are supposed to embody and exhibit the oxymoron of something both unending and complete.25placeholder
The same Leibniz who, as I said, was jointly responsible for the discovery of the calculus, where infinities circulate freely, also expressed qualms about infinity:
“[W]e believe that we are thinking of many things (though confusedly) which nevertheless imply a contradiction; for example, the number of all numbers. We ought strongly to suspect the concepts of infinity, of maximum and minimum, of the most perfect, and of allness itself.”26placeholder
To corroborate this apprehensive cautiousness, the set theory Cantor developed by giving free rein to actual infinities soon became the theater of proliferating antinomies, and had to be revised and limited in its range by suitable axioms. The current state of the art, in light of Gödel’s theorems, is that the relevant formal systems have not yet produced additional antinomies, but it is impossible to prove that they will never do so. We can leave such momentous affairs aside, however, as actual infinities don’t enter our daily dealings with √2; what we confront there is rather, again, an approximation internal to mathematics—that of a decimal expansion which never stops and we can only ever use an initial portion of. We confront an increasing empirical familiarity with √2 (similar to the familiarity we could gain, increasingly, with a neighbor by getting to know more and more details about him) which never graduates into a priori knowledge of it (as the familiarity with a neighbor would never graduate into perfect, unshakeable knowledge of him).
There are many more irrational numbers than rational ones; indeed, the irrationals are a higher order of infinity than the rationals. The irrationals like √2, however, are not more than the rationals: they are called algebraic because they are solutions of algebraic equations (√2 is the positive solution of the equation x2 = 2) and we can prove that there are as many algebraic numbers as there are rationals. What makes the cardinality of irrationals exceed (by a whole lot) that of rationals are transcendental numbers, which are not solutions of algebraic equations. Of an irrational algebraic number we don’t know the identity (if we avoid trafficking with actual infinities) and we can only approximate it, but at least we can name it: we can state a property that uniquely describes it—we can say that √2 is “the solution of the equation x2 = 2,”27placeholder as we could say that “the Cat” uniquely describes an unknown, acrobatic jewel thief.28placeholder For a transcendental number we can provide no such name: not only is its identity unknown but also any property that would describe it uniquely. We literally don’t know what we are talking about.
The most important number in all of mathematics is transcendental. It is π,29placeholder which should designate the ratio between the circumference and the diameter of a circle (but we will see that this characterization is way too simple), and we find it everywhere. Since in the previous section I discussed trigonometry, I might at least cite the measurement of angles in radians, used by the International System of Units: with reference to the following figure, a radian is the ratio between the length of the arc corresponding to the angle (understood as a central angle) and the length of the relevant radius (the radius r is half the diameter). By this unit of measurement, every angle turns out to be either a fraction or a multiple of π; but what is π?
Say that you use some instrument to measure the circumference and the diameter of a circle, and obtain numbers a and b, respectively, of whatever finite complexity. These measurements must necessarily be wrong, for otherwise π would be the fraction a/b and would be a rational number! Therefore, when I spoke above of the ratio between the circumference and the diameter of a circle I was speaking of magnitudes at least one of which was itself transcendental and could not be exactly measured or calculated—or uniquely described. I was speaking intuitively of something that cannot be made mathematically precise. It is as if I was speaking of the ratio between God’s power (or wisdom, or goodness) and that of humans: in that case, too, one term escapes our rational control and can only be understood by (weak) analogy. At first I might think that I was speaking of something like the ratio between the power (or wisdom, or goodness) of a human X and that of another human Y, but No: I would be dealing with something different—radically, even incomprehensibly, different.30placeholder The term “transcendental,” it seems, and the corresponding quality of transcendence, were not chosen at random: they are highly suggestive of what is going on here.31placeholder
So, again, what is π? An unknowable, unnameable number we try to approximate by various calculations. One is the following: If the circumference of a circle is 2πr, its area C is πr2. Say that r = 10, inscribe the circle inside a square and fill the square (whose area S = 400) with little squares of side 1. We can fit 277 inside the circle, while 338 will not be entirely outside of it, with an average of 307.5. Since the area S of the square is 4r2, C/S = π/4, and π = 4C/S. Putting 277, 307.5, and 338 for C, we get 2.77, 3.075, and 3.38, respectively; then the average gives us π = 3.075, with an error (corresponding to half the difference between 2.77 and 3.38) estimated as 0.305—a first approximation to π. A rudimentary approximation, since by adding squares we cannot obtain a circle: that it is impossible to “square the circle” is a consequence of π being transcendental, hence the former was proved by Ferdinand von Lindemann in 1882 by proving the latter.32placeholder But we can get better approximations, by extending the length of r and thus multiplying the number of squares involved, hence reducing the difference between their total area and the area of the circle. This is a stage of the procedure, with green and yellow, respectively, for the little squares totally inside and not totally outside the circle:
And this is an initial portion of its outcome (notice how the error gets reduced):
| r | approximation of π |
| 15 | 3,104, with an error estimated as 0.22 |
| 20 | 3,251, with an error estimated as 0.166. |
The ideal destination of this journey would be squares that reduce to simple points, but we will never get there. What we can obtain is a difference between the area of the circle and the total area of the squares that becomes always more imperceptible. Leonardo, who was obsessed by squaring the circle and twice claimed to have achieved it, in another passage writes more reasonably that he had attained not a result which was in fact out of reach but rather one which was at an imperceptible distance from the one out of reach. In the Anatomic Drawings from the Royal Library, Windsor Castle we read:
“And I square the circle minus a portion as small as the intellect can imagine, that is, as the visible point.”33placeholder
By having multiple computers work jointly, today we can calculate π with a humongous number of decimal digits;34placeholder but that is still at an infinite distance from what π is. As would be a believer who, by increasingly more demanding spiritual and mystic experiences, reached a better idea of God than anyone else—it would still be a finite idea, situated at an infinite distance from the object of his search.
It is important to clarify what the above shows and what it does not. That the area of a circle with radius r be πr2 is an eternal mathematical truth that can be known a priori. In this respect, it is no different from the eternal truths that certain configurations on a chessboard amount to checkmate—those, too, are independent from the relevant chess games being empirically played, or from chess even being practiced in the empirical world. But the a priori status of such truths, for both kinds, does not extend to what we can learn from them about Humean “matters of fact.” If we want to assign a numerical value to the area of an empirical circle, we can measure its radius and then start the calculation of π, which is an empirical procedure and also one that will never come to an end. Therefore, there will never be any necessity to a concrete number that, at any stage of this procedure, we assign to the area of the circle: it will always be the empirical outcome of an empirical operation. Even worse, we will always know that any such number (as was the case for the sine of 28.5°, and for √2) is wrong—this much, indeed, is something we can prove.
4. Limits
Suppose I deposit $1 in a bank, at an interest of 100%, and that the interest is only credited to my account at the end of the year. At that time I will have $2 deposited, and the following year the interest will be calculated on the larger amount, eventually giving me $4. This is what is known as the compounding of interest: every time interest is credited, it is so not only on the original deposit but also on any interest that has matured in the meantime. It makes a difference how often interest is credited: if, in my case, I was credited interest not at the end of the year but every six months, I would have $1.5 after the first six months and $2.25 at the end of the first year; if it was every three months, I would have in excess of $2.44 at the end of the first year (this “in excess” will soon become clear); and, if it was every day, I would have in excess of $2.71 at the end of the first year. What would happen, however, if I was credited interest continuously—that is, an infinite number of times during a year (at every instant, we could say)? In 1683 the Swiss Jacob Bernouilli (a member of a family which counted no less than eight renowned mathematicians) addressed this question in the context of the newly discovered calculus (in the diatribe on its discovery between Newton and Leibniz, he took the latter’s side), using what is probably the calculus’ most fundamental notion: that of a limit.
A mathematical function like x + 3 has different values for different arguments: for the argument 2 the value is 5, and for the argument 7 the value is 10. When studying a limit of a function we are not looking at a particular argument (or value), but rather at what happens to the value around a certain argument, as the argument approaches, or tends to (but does not straightforwardly coincide with) a certain magnitude. For example, what happens to the value of the function x + 3 as x gets closer and closer to 10? Is there a value L that we can get arbitrarily close to, as long as the argument x gets close enough to 10? If there is, then L is the limit of x + 3 for x approaching 10; in symbols,
\(\lim\limits_{x \to 10} (x + 3) = L.\)
(In fact, this case is trivial because L is just 10 + 3 = 13; but not all cases are so, including the one I am about to turn to.)
Bernouilli proved that continuous compounding would produce an amount that depended (in addition to the principal, the rate, and the time) on the limit of a certain function for its argument approaching infinity. Specifically, if n is the number of times interest is credited in a year and ∞ is the symbol for infinity, what he found himself looking for was the following limit:
\(\lim\limits_{n \to \infty}(1 + 1/n)^{n}\)
This limit was later called e by Leonhard Euler and is, next to π and possibly on a par with it, the most important number in mathematics; among other things, it is the base of natural logarithms. It is known as Euler’s number, or as Napier’s constant after John Napier—who discovered logarithms but not e. It is also, like π, a transcendental number; therefore it cannot be written and it cannot be named (identified by some unique mathematical property). In case you are interested, the first thirty places of its infinite decimal expansion are
2.718281828459045235360287471352.
Instead of writing or naming it, we can calculate it, as we did with π, and with a similarly inconclusive outcome. What we could do is add to one another all members of the following series:
1, 1/1, 1/1×2, 1/1×2×3, 1/1×2×3×4, 1/1×2×3×4×5, …
For illustration, here are some initial (approximate) values of this calculation:
1
2
2.5
2.666
2.707
2.715
Clearly, these values approach the thirty-place expansion written above; and, just as clearly, they will never give us anything definite (as didn’t the previous calculation of compound interest, that after a while could only be said to be “in excess of” certain values). Which I take the liberty of generalizing to the whole notion of a limit: it is a great conquest of the human mind, and tremendously useful for greatly many purposes; but, from the point of view I am taking here, when a limit is not as trivial as in my first example, it often issues not in a number but in the infinite search for a number—in an infinite procedure of calculation. An empirical procedure of calculation, whose results are obtained one by one (as is the case with π) by performing empirical operations—and are always (as with π) inadequate to the object of the search.
5. Integrals
Time to take up Galileo’s challenge for good. It is not only perfect shapes (perfect spheres, say) that “abstract” mathematics can account for, he claimed, but all those that present themselves to us in concrete, ordinary life, however irregular they might be. The calculus addresses and (apparently, I said) meets this challenge, by using integrals. Let us see how that works; of the numerous variants available I will consider the most common one, due to Bernhard Riemann, and for simplicity I will just deal with definite integrals. The task is calculating the area of any surface whatever; for example, with reference to the figure below, measuring the area identified by the curve (corresponding to some function f(x)) and by the two straight lines, parallel to the y Cartesian axis, drawn from points a and b.35placeholder
In analogy with what we did when calculating π, let us approximate the desired area by covering it with rectangles of decreasing width, as follows:
As the figure shows, the narrower the rectangles are, the more their total area approaches the desired one. Ideally, if the rectangles’ width were reduced to zero—if the rectangles had no width—their total area would coincide with the desired one. So let us introduce the limit of this operation of refinement: the value the total area of the rectangles tends to as their width approaches zero. That is the integral of the function, and it gives the desired area. The symbol for the integral was introduced by Leibniz and reminds one of the symbol for a sum: whereas the sum of a number of values is represented by the Greek sigma ∑, for the integral Leibniz chose a long sigma ∫; so the complete notation is \(\int_{a}^{b}f(x)\,dx\), to be read “the infinite sum of the values of f(x) from a to b for infinitesimal intervals dx.”
Have we thus accomplished the miracle of calculating the area of any surface whatever? Once more, it depends on how you look at it. Consider an example. Here the function is √x and the rectangles give us situations like the following (the green ones smaller and the yellow ones larger than the desired area):
And at the limit? If what we want is (as before) the definite integral for an area identified by the curve and two parallel lines to the y axis drawn from points a and b on the x axis, some routine calculations give us the value 2/3(√b3 – √a3), where the presence of the square roots of a and b to the third power involves us with irrational numbers, hence with (additional) infinite, not-completable calculations. No problem naming the solution; but no hope of ever writing it!
For a second example take the function 1/x, whose graphical representation on Cartesian axes is a hyperbola:
Now suppose we want to calculate areas identified by this curve, the x axis, and parallels drawn to the y axis from the points 1, 3, and 6 on the x axis. Again, routine calculations give us, for the relevant definite integrals, ln(3) from 1 to 3, ln(2) from 3 to 6, and ln(6) from 1 to 6, respectively (ln is the symbol for the natural logarithm). All these values are transcendental, as are all natural logarithms of non-zero algebraic numbers other than 1; so these areas not only cannot be written, they cannot even be named. (And don’t be fooled by the locution ln(x). For the cases I considered, that is, literally, hand waving: it no more names a definite number than “God’s plan” names a definite plan or an arrow at the side of a road names the end of the road.) Galileo’s accountant would be in big trouble if, after discounting all the boxes, bales, and other packings here, he had to say how much sugar, silk, or wool he had left.
Conclusion
Mathematical proofs yield a priori knowledge. In this sense, it is true that mathematics is not an opinion: it has the necessity and universality Kant and many others have claimed for it. In this sense, however, it is no different from any other abstract game, where analogous proofs can be carried out, providing knowledge that is equally a priori—equally universal and necessary. As I said before, the mathematical game has been played with greater intensity, continuity, and creativity than any other; so the range of a priori truths it provides is greater than what any other game could offer. Years ago Dan Dennett thought of a game similar to chess, which he called chmess, where the king could move two cases, and correctly said that this other game could have as much profundity, and generate problems that were quite as hard, as regular chess.36placeholder I can imagine chmess being regarded not as a game distinct from chess, but as a member of a family of chess-like games, to be explored in all their amazingly subtle variations and complexities—much as the rational numbers of the ancient Greeks were later recognized as members of a larger family that included irrational, transcendental, and imaginary cousins. And I cannot imagine how far we could go by playing that family of chess relatives with the greatest intensity, continuity, and creativity. One thing I can say for sure: the range of a priori truths thus obtained would be much much larger than it is now, for the sort of chess we currently practice.
When a game offers a repertory of a priori truths, it can be used as a model against which to assess reality. In the case of chess, I could take the king to model a real political leader, the queen his most influential advisor, the bishops his ambassadors, the knights his generals, the rooks his riches, and the pawns his citizens, and try to get enlightened about a real-life situation by playing a game. But of course the king of chess is not a real political leader, and the same is true for all other pieces; therefore, I will only get limited enlightenment from such modeling, which I will have to supplement with the sort of savvy I have (hopefully) acquired by practice with real environments and real events. Abstract modeling will only go so far; at some point phrónesis will have to come in.
Because of the intensity, the continuity, and the creativity with which it has been played, mathematics has gained an extraordinary status, and an opinion has been circulating that its a priori truths apply directly to the empirical world. I have argued here that it is a wrong opinion. If we restrict ourselves to the cases for which mathematics can actually deliver a priori verdicts, we are going to confirm what Plato and Hume believed: those verdicts can only be approximated by the real objects we encounter in experience. Just like the verdicts delivered by any other abstract game. If we try to make mathematics more directly responsive to the real structure of the empirical world, we also make it prey to the same approximations: its a priori verdicts are going to be replaced by empirical procedures of calculation—like those executed by computers that induced doubts on mathematics’ a priori character, but with much less efficiency, when performed by humans, than what computers can attain. Machines are much better than humans at calculating infinite algorithms; still, those algorithms are not, and are never going to become, proofs, and are never going to provide the definite (and definitive) outcomes we expect from proofs. QED.
Bottom line: Enjoy mathematics if that is a game you like playing; the more you play it and enjoy it, and consequently expand it and deepen it, the more correspondences you will be able to establish with real-life situations. But don’t think that those correspondences can give you the truth of real-life situations. That is the wrong opinion to have. What you will ever get is estimates of the truth. And you may always end up with as much of a disaster on your hands at the chess grandmaster Tov Kronsteen.
Quoted in The Guarded Gate, by Daniel Okrent (New York: Scribner, 2019), 257.
I thank Fausto Di Biase for his useful comments on a previous draft of this paper. As we remain in sharp disagreement on my line of argument, I exempt him from any responsibility for it.
For a highly representative example of this attitude, see Kant’s second preface to his Critique of Pure Reason: “Mathematics has, from the earliest times to which the history of human reason reaches, in that admirable people the Greeks, traveled the secure path of a science. […] In metaphysics we have to retrace our path countless times, because we find that it does not lead where we want to go, and it is so far from reaching unanimity in the assertions of its adherents that it is rather a battlefield” (translated by Paul Guyer and Allen W. Wood; Cambridge UK: Cambridge University Press, 1998; 107-109, Bx-Bxv).
Kant, Critique of Pure Reason, (translated by Paul Guyer and Allen W. Wood; Cambridge UK: Cambridge University Press, 1998; 127, A2).
Kant, Critique of Pure Reason, (translated by Paul Guyer and Allen W. Wood; Cambridge UK: Cambridge University Press, 1998; 129, A4|B8).
We know at least since the discovery of non-Euclidean geometries, to be mentioned later (Plato, also mentioned later, knew it long before that), that the notions of a priori, universality, and necessity referred to here must not be taken the way Kant did. For we know now that (in contrast with what he believed) there are several, equally viable mathematical theories, based on different, conflicting axioms, and they cannot all be a priori, universal, and necessary in any absolute sense. It is, rather, that no such theories are to be verified or falsified empirically, as, say, physical theories are according to the hypothetico-deductive model. They could only be falsified by showing an internal contradiction in them, not by showing them to contradict some empirical evidence; it is in this sense that I regard them as “prior to, and independent of, any experience.” (It is reported that, when General Relativity was confirmed by observations during a Solar eclipse in 1919 and Einstein was asked about the possibility of it not being confirmed, he answered: “I would have had to pity our dear God. The theory is correct all the same” (see it in Reality and Scientific Truth: Discussions with Einstein, von Laue, and Planck (Detroit MI: Wayne State University Press, 1980), 74, by Ilse Rosenthal-Schneider). One way of understanding this anecdote is that, as a mathematical theory, General Relativity was immune from falsification by empirical observation—and its beauty and power would have made this world a worse one if it did not go along with it—whereas of course, as a physical theory, such immunity was not to hold for it.) See also the following footnote.
Some philosophers (a classic example is John Stuart Mill, a more recent one Willard Van Orman Quine) do not share this received view, and take mathematics to be empirical. Some mathematicians are also similarly inclined. Vladimir Igorevich Arnold begins his “On Teaching Mathematics” (Russian Mathematical Surveys 53 (1998), 229) saying: “Mathematics is a part of physics. Physics is an experimental science. Mathematics is the part of physics where experiments are cheap.” While these authors are not my target here, hence I don’t take issue with them, I believe they are somewhat simpleminded, and are throwing away the baby with the bathwater. Mathematics has an important a priori component, as any other abstract game (for which see below); that component, however, does not reach nearly as far as the received view would have it.
In Terence Young’s movie From Russia with Love (1963) an intricate plan for trapping James Bond and having him steal a cryptography device from the Soviets on behalf of Spectre is hatched by the Czechoslovak chess grandmaster Tov Kronsteen (played by the Polish Vladek Sheybal). But the plan has a disastrous outcome, and Kronsteen is executed by Spectre’s chairman Ernst Stavro Blofeld.
See Diogenes’ The Lives and Opinions of Eminent Philosophers, translated by Charles Duke Yonge (London: G. Bell and Sons, 1915), 343.
See his Process and Reality: An Essay in Cosmology, edited by David Ray Griffin and Donald W. Sherburne (New York: The Free Press, 1978), 39.
Edited by Peter Millican (Oxford UK: Oxford University Press, 2007), 18.
See his Ideas and Opinions (New York: Crown Publishers, 1954), 271.
See his A Brief History of Time: From the Big Bang to Black Holes (New York: Bantam, 1988), 179.
Translated by Stillman Drake (Berkeley and Los Angeles: University of California Press, 1967).
The Inquisition condemned this statement. But the Inquisition’s God was soon to become irrelevant, and Galileo’s proud claim was to stand.
Plato was fascinated by these solids (also referred to, at times, as Platonic solids), and pores over them at length in the Timaeus, 53cff. See it in Alfred Edward Taylor’s translation of Plato: Timaeus and Critias (London: Methuen, 1929), 52ff.
When I talk about decimal expansions that never end, here and below, I refer to expansions that (besides being infinite) never settle into a repeating pattern—as, say, 0.3333… or 0.5144144144… do. For those repeating (also called “recurring”) decimals are in fact rational numbers (for which, again, see the next section); 0.3333…, for example, is the quotient of 1 and 3, hence corresponds to the rational number 1/3.
On the Most Ancient Wisdom of the Italians Unearthed from the Origins of the Latin Language, by Giambattista Vico, translated by Lucia M. Palmer (Ithaca NY: Cornell University Press, 1988), 181.
In the entry on philosophy of mathematics from the Stanford Encyclopedia of Philosophy, Horsten says that “when we rely on a computer to generate part of a proof, we appear to rely on the proper functioning of computer hardware and on the correctness of a computer program,” and that “[t]hese appear to be empirical factors.” But, when similar calculations are performed by a human (whether or not he is aided by a computer), are we not relying on the proper functioning of his brain?
Aristotle mentions this proof in Prior Analytics I 23, without much detail. All he says is: “one proves that the diagonal of a square is incommensurable [i. e., that it has no common, rational unit of measurement] with the sides by showing that if it is assumed to be commensurable, odd become equal to even numbers” (translated by Hugh Tredennick in The Categories On Interpretation Prior Analytics; London: William Heinemann, 1962, 321). A detailed proof (different from the one above, though still by reductio) is present in Euclid’s Elements, Book X, Proposition 117; but it is regarded as an interpolation.
See the original at www.leonardodigitale.com; translation mine.
Braunschweig: Friedrich Vieweg und Sohn, 1872.
I have addressed this notion in “In a Manner of Speaking: A Vindication of Potential Infinity”, Epoché Magazine, 60 (2023).
Actual infinities were defended by Hegel in his Science of Logic of 1812/1816 (translated by George di Giovanni; Cambridge UK: Cambridge University Press, 2015), two generations before their gaining dominance in set theory. Since Kant favors thinking of infinity as endless progress, the contrast between potential and actual infinity can be seen, philosophically, as one between Kant and Hegel.
Gottfried Wilhelm Leibniz, Philosophical Papers and Letters, edited by Leroy Earl Loemker (Dordrecht: Reidel, 1969), 166. This passage comes from a letter written by Leibniz to his correspondent Henry Oldenburg and dated December 28, 1675.
As a matter of fact, there are two such solutions, a positive and a negative; but we can handle that complication (as I did above) by uniquely identifying them as “the positive solution of the equation x2 = 2” and “the negative solution of the equation x2 = 2,” respectively.
This example is borrowed from Alfred Hitchcock’s movie To Catch a Thief (1955), where the Cat is played by Cary Grant. (Curiously, at the end of the movie there are shown to be both a real Cat and an imitation one; so that description, too, might need to be disambiguated.)
Another possible candidate for this exalted role will emerge later. And it will be transcendental, too.
Notice that what is transcendental is the ratio π between circumference and diameter. What numbers we actually (try to) assign to both will depend on our unit of measurement. If the diameter is the unit of measurement then the circumference is transcendental and, if it is the circumference, then the diameter is transcendental.
The Latin “transcendens” was first applied to this context by Leibniz when investigating representations of π in his “De vera proportione circuli ad quadratum circumscriptum in numeris rationalibus” (Acta eruditorum 1682, 41-46). See it in a 2014 English translation by Ian Bruce at https://www.17centurymaths.com/contents/Leibniz/ae6.pdf.
“Über die Zahl π.” Mathematische Annalen 20 (1882), 213-225.
See the original at www.leonardodigitale.com; translation mine. A visible point for Leonardo is not a mathematical one, but is rather the “point” we can leave on paper by using a pen.
In March 2022 Emma Haruka Iwao used Google Cloud to calculate 100 trillion digits of π.
This task cannot be fulfilled in all cases. It cannot be when the function has “too many” (or, rather, the wrong kind of) discontinuities. But I disregard such complications here.
“Higher-Order Truths about Chmess.” Topoi 25 (2006), 39-41.