enly or not, conceptualizing infinity as a number, then it is our job as cognitive scientists to characterize the cognitive mechanism by which they are making that "mistake." And if we are correct in suggesting that a single cognitive mechanism-namely, the BMI-is used for all conceptions of infinity, then we have to show how the BMI can be used to conceptualize infinity as a number, whether it is a "mistake" to do so or not. Cognitive science is, after all, descriptive, not prescriptive. And it must ex- plain, as well, why people think as they do. The "mistake" of thinking of in- finity as a number is not random. "oo" is usually used with a precise meaning-as a number in an enumeration, not as a number in a calculation. In "1, 2, 3, . . . , oo" ooo is taken as an endpoint in an enumeration, larger than any finite number and beyond all of them. But people do not use ooo as a number in calculations: We see no cases of "17 times oo, minus 473," which is of course a meaningless expression, as Hardy correctly points out. The moral here is that there are, cognitively, different uses for numbers-enumeration, comparison, and calculation. As a number, ooo is used in enumeration and comparison but not in calculation. Even mathematicians use infinity as a number in enumeration, as in the sum of a sequence an from n = 1 to n = oo: When Hardy warns us not to assume that ooo is a number, it is because mathe- maticians have devised notions and ways of thinking, talking, and writing, in which ooo is a number with respect to enumeration, though not calculation. Indeed, the idea of ooo as a number can also be seen as a special case of the BMI. Note that the BMI does not have any numbers in it. Suppose we apply the BMI to the integers used to indicate order of enumeration. The inherent structure of the target domain, independent of the metaphor, has a potential infinity, an un- ending sequence of ordered integers. The effect of the BMI is to turn this into an actual infinity with a largest "number" oo. The BMI itself has no numbers. However, the unending sequence of integers used for enumeration (but not calculation) can be a special case of the target do- main of the BMI. As such, the BMI produces ooo as the largest integer used for enumeration. This is the way most people understand ooo as a number. It cannot be used for calculation. It functions exclusively as an extremity. Its operations are largely undefined. Thus, oo/0 is undefined, as is ooo * 0, ooo - oo, and oo/oo. And thus ooo is not a full-fledged number, which was Hardy's point. For Hardy, an en- tity either was a number or it wasn't, since he believed that numbers were ob- jectively existing entities. The idea of a "number" that had one of the functions of a number (enumeration) but not other functions (e.g., calculation) was an im- possibility for him. But it is not an impossibility from a cognitive perspective, and indeed people do use it. ooo as the extreme natural number is commonly used with the implicit or explicit sequence "1, 2, 3, . . . , oo in the characterization of infinite processes. Each such use involves a hidden use of the BMI to con- ceptualize oo as the extreme natural number. Mathematicians use "1, .... n, . . ., oo" to indicate the terms of an infinite sequence. Although the BMI in itself is number-free, it will be notationally con- venient from here on to index the elements of the target domain of the BMI with the integers "ending" with metaphorical "oo." For mathematical purists we should note that when an expression like n-1 is used, it is taken only as a notation indexing the stage previous to stage n, and not as the result of the op- eration of subtraction of 1 from n. Projective Geometry: Where Parallel Lines Meet at Infinity In projective geometry, there is an axiom that all parallel lines meet at infinity. >From a cognitive perspective, this axiom presents the following problems. (1) How can we conceptualize what it means for there to be a point "at infinity?" (2) How can we conceptualize parallel lines as "meeting" at such a point? (3) How can such a conceptualization use the same general mechanism for com- prehending infinity that is used for other concepts involving infinity? We will answer these questions by taking the BMI and filling it in in an ap- propriate way, with a fully comprehensible process that will produce a final re- sult, notated by "oo," in which parallel lines meet at a point at infinity. Our answer must fit an important constraint. The point at infinity must function like any other point; for example, it must be able to function as the intersection of lines and as the vertex of a triangle. Theorems about intersections of lines and vertices of triangles must hold of "points at infinity." To produce such a special case of the BMI, we have to specify certain parameters: • A subject-matter frame, indicating that the subject matter is geometry- specifically, that it is about lines that intersect at a point. • The initial step of the process. • The iterated process, or a step-by-step condition that links each state resulting from the process to the next state. • The resultant state after each iteration. • An entailment of the uniqueness of the final resultant state. In the case of projective geometry, we take as the subject-matter frame an isosceles triangle, for reasons that will shortly become clear (see Figure 8.2). The iterative process in this case is to move point Cn further and further away from points A and B. As the distance Dn between A and Cn gets larger, the an- gles an and bn approach 90 degrees more and more closely. As a result, the in- tersecting lines Lln and L2n, get closer and closer to being parallel. This is an unending, infinite process. At each stage n, the lines meet at point Cn. (For de- tails, see Kline, 1962, ch. 11; Maor, 1987.) The idea of "moving point Cn further and further" is captured by a sequence of moves, with each move extending over an arbitrary distance. By the condi- tion "arbitrary distance" we mean to quantify over all the distances for which the condition holds. Here are the details for filling in the parameters in the BMI in this special case. As a result of the BMI, lines L1 and L2 are parallel, meet at infinity, and are sep- arated by the length of line segment AB. Since the length AB was left unspeci- fied, as was the orientation of the triangle, this result will "fit" all lines parallel to L1 and L2 in the plane. Thus, this application of the BMI defines the same sys- tem of geometry as the basic axiom of projective geometry-namely, that all parallel lines in the plane meet at infinity. Thus, for each orientation there is an infinite family of parallel lines, all meeting "at infinity." Since there is such a family of parallel lines for each orientation in the plane, there is a "point at in- finity" in every direction. Thus, we have seen that there is a special case of BMI that defines the notion "point at infinity" in projective geometry, which is a special case of actual in- finity as defined by the BMI. We should remind the reader here that this is a cog- nitive analysis of the concept "point at infinity" in projective geometry. It is not a mathematical analysis, is not meant to be one, and should not be confused with one. Our claim is a cognitive claim: The concept "point at infinity" in pro- jective geometry is, from a cognitive perspective, a special case of the general notion of actual infinity. We have at present no experimental evidence to back up this claim. In order to show that the claim is a plausible one, we will have to show that a wide va- riety of concepts of infinity in mathematics arise as special cases of the BMI. Even then, this will not prove empirically that they are; it will, however, make the claim highly plausible. The significance of this claim is not only that there is a single general cogni- tive mechanism underlying all human conceptualizations of infinity in mathe- matics, but also that this single mechanism makes use of common elements of human cognition-aspect and conceptual metaphor. The Point at Infinity in Inversive Geometry Inversive geometry also has a concept of a "point at infinity," but it is a concept very different from the one found in projective geometry. Inversive geometry is defined by a certain transformation on the Cartesian plane. Consider the Carte- sian plane described in polar coordinates, in which every point is represented by (r, 0) where 0 is an angle and r is the distance from the origin. Consider the trans- formation that maps r onto 1/r. This transformation maps the unit circle onto itself, the interior of the unit circle onto its exterior, and its exterior onto its in- terior. Let us consider what happens to zero under this transformation. Consider a ray from zero extending outward indefinitely at some angle 0 (see Figure 8.3). As r inside the circle gets closer to 0, 1/r gets further away. Thus, 1/1,000 is mapped onto 1,000, 1/1,000,000 is mapped onto 1,000,000, and so on. As r approaches 0, I/r approaches oo. What is the point at 0 mapped onto? It is tempting to map 0, line by line, into a point at ooo on that line. But there would be many such points, one for each line. What inversive geometry does is define a single "point at infinity" for all lines, and it maps 0, which is unique, onto the unique "point at infinity." If we are correct that the BMI characterizes actual infinity in all of its many forms, then the point at infinity in inversive geometry should also be a special case of the BMI. But, as in most cases, the precise formulation takes some care. Whereas in projective geometry there is an infinity of points at infinity, in in- versive geometry there is only one. This must emerge as an entailment of the BMI, given the appropriate parameters: the frame and the iterative process. Here is the frame, which fills in the "subject matter" parameter of the BMI. The frame is straightforward. THE INVERsivE GEOMETRY FRAME The Cartesian plane, with polar coordinates. The one-to-one function fix) = 11x, for x on a ray that extends from the origin outward. This frame picks a special case of the target domain of the BMI. To fill out this special case of the BMI, we need to characterize the iterative process. The idea behind the process is simple. Pick a ray and pick a point xl on that ray smaller than 1. That point has an inverse, which is greater than 1. Keep picking points xn on the same ray, closer and closer to zero. The inverse points 1/xn get further and further from the origin. This is an infinite, unending process. We take this process as the iterative process in the BMI. Here are the details: The BMI then applies to this infinite, unending process and conceptualizes it metaphorically as having a unique final resultant state "oo." At this metaphori- cal final state, x is at the origin, and the distance from the origin to xoo' is infi- nitely long. In inversive geometry, arithmetic is extended to include oo as a num- ber with respect to division: D = oo, 1/Doo = 1/oo = 0, and 1/0 = oo. This extension of arithmetic says nothing about the addition or subtraction of oo. It is peculiar to inversive geometry because of the way this special case of the BMI is defined: All points must have inverses, including the point at the origin. Where dividing by zero is normally not possible, this metaphor extends ordinary division to give a metaphorical value to 1/0 and 1/oo. In inversive geometry, oo does exist as a number and has limited specified possibilities for calculation. Moreover, since there is only one zero point shared by all rays, and since f(x) = 1/x is a one-to-one mapping that, via this metaphor, maps 0 onto ooo and onto 0, there must therefore be only one ooo point shared by all rays. What is interesting about this case is that the same general metaphor, the BMI, produces a concept of the point at infinity very different from that in the case of projective geometry. In projective geometry, there is an infinity of points at infinity (for an image, think of the horizon line), while in inversive geometry there is only one. Moreover, projective geometry has no implicit associated arithmetic, while inversive geometry has an implicit arithmetic (differing from normal arithmetic in its treatment of zero and infinity). Finally, though we have characterized inversive geometry in terms of a cog- nitive mechanism, the BMI, mathematicians of course do not. They simply de- fine inversive geometry in normal mathematical terms. Our goal is to show how the same concept of infinity is involved in inversive geometry and other forms of mathematics, while respecting the differences in the concept of infin- ity across branches of mathematics. The Infinite Set of Natural Numbers Within formal arithmetic, the natural numbers are usually characterized by the successor operation: Start with 1. Add 1 to yield a result. Add 1 to the result to yield a new result. And so on. This is an unending, infinite process. It yields the natural numbers, one at a time. Not the infinite set containing all the natural numbers. Just the natural numbers themselves, each of which is finite. Since it is incapable of being used in calculation, ooo is not a full-fledged member of the infinite set of natural numbers. Here is the problem of characterizing the set of natural numbers. The set must be infinite since it contains all of the infinitely many numbers, but it can- not contain ooo as a number. To get the set of natural numbers, you have to collect up each number as it is formed. The set keeps growing without end. To get the entire set of natural numbers-all of them, even though the set never stops growing-you need something more. In axiomatic set theory, you add an axiom that simply stipu- lates that the set exists. From a cognitive perspective, that set can be con- structed conceptually via a version of the Basic Metaphor of Infinity. The BMI imposes a metaphorical completion to the unending process of natural-number collection. The result is the entire collection, the set of all natural numbers! This special case of the BMI does the same work from a cognitive perspective as the axiom of Infinity in set theory; that is, it ensures the existence of an in- finite set whose members are all the natural numbers. This is an important point. The BMI, as we shall see, is often the conceptual equivalent of some axiom that guarantees the existence of some kind of infinite entity (e.g., a least upper bound. And just as axioms do, the special cases of the BMI determines the right set of inferences required.