TY - JOUR
AU - Reeder, Patrick
AB - ABSTRACT This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between (i) the conceptions that treat the whole continuum as prior to its parts, and (ii) those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity (i) and those that are more readily available for mathematico-scientific use (ii). There are two labyrinths of the human mind: one concerns the composition of the continuum, and the other the nature of freedom, and both spring from the same source — the infinite. ‘On Freedom’ [Leibniz, 1956, pp. 406–407] The concept of continuity has many incarnations in contemporary mathematics. As a first gloss, something is continuous if it is represented naturally by a line, plane, or solid. This gloss suggests that continuity is a geometric concept at its core. Many of the continuum’s philosophically intuitive features are not naturally accommodated by modern mathematical machinery. Numerous attempts have been made to explicate continuity. These attempts at explication have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. Indeed, labyrinthine obstacles thwart earnest efforts to provide a continuum that is both philosophically sensitive and mathematically useful. One might wonder, What do these different conceptions of continuity have in common? Is there not some thread that ties them all together? In what follows, I argue that all of these conceptions satisfy: (1) Infinite Divisibility. An object $$X$$ is infinitely divisible if and only if $$X$$ is (a) extended and (b) for any bounded, extended part $$Y$$ of $$X$$, $$Y$$ can be bisected along some axis ($$2^n-1$$ times) into $$2^n$$ extended parts, for any positive integer $$n$$.1 As I will demonstrate more below, there are mathematical objects that satisfy Infinite Divisibility but are not continuous. Some further principle must be added. It is here that any would-be thread unravels: (2a) Punctilious Points. An object $$X$$ is continuous if and only if $$X$$ is composed of a linear ordering of points and for any partition $$A,B$$ of $$X$$ such that all members of $$A$$ are less than all members of $$B$$, there is exactly one point $$c$$ that is either the greatest member of $$A$$ or the least of $$B$$. (2b) Intransigent Infinitesimals. An object is continuous only if it is composed of infinitesimal magnitudes. (2c) Atomless Gunk. An object is continuous only if all of its parts have a proper part. (2d) Intuitionistic Indecomposables. An object is continuous only if it is not identical to a sum of any of its disjoint proper parts. (2e) Prodigious Possibilities. An object $$X$$ is continuous only if for any cardinal number $$\kappa$$ and for any two distinct points on $$X$$, there are at least $$\kappa$$ parts between those points. Although some of the above requirements are logically compatible with others, each of these requirements has been selected so as to highlight a unique conception of the continuum. The reader should not assume that any one of these requirements conjoined with Infinite Divisibility would be sufficient to form a continuum. Rather, for each conception, the associated requirement was selected to represent the most philosophically striking feature of that conception. Furthermore, although the concept of continuity has its roots in history and its flower in mathematics, this is not a direct survey of either the history or the mathematics involved. This is a conceptual survey with historical and mathematical highlights. Here is a roadmap. First, I will briefly examine the basic intuitions and ideas behind Infinite Divisibility, highlighting why it is a necessary but insufficient criterion. Second, I will offer a novel categorization of the five conceptions. After that, I will unravel each of the above requirements (2a)–(2e) one by one. For those less familiar cases, I will offer brief expositions of their contemporary mathematical expressions. Throughout, a strong tension will emerge between (i) fidelity to core philosophical presuppositions about the continuum, and (ii) the practical, scientific needs of working mathematicians: no conception will satisfy both our philosophical and mathematico-scientific appetites. 1. INFINITE DIVISIBILITY Why are continuous phenomena infinitely divisible? It is helpful to compare continuity with discreteness. Aristotle writes: Of quantities some are discrete, others continuous; and some are composed of parts which have position in relation to each other, others are not composed of parts which have position. Discrete are number and language; continuous are lines, surfaces, bodies and also, besides these, time and place. (Categories VI, 4b20–25) Discrete things are marked by their separateness. Some discrete things might together compose a whole, but even then the whole can be separated into natural, distinguishable parts. 2 He offers the syllables of spoken language as such an example. For each word, it can be broken into its component syllables naturally. Each syllable is ultimately distinguishable, even though within a given word they are in some sense together. In contrast, continuous things form a homogeneous unity. The parts of continuous things qua continuous are in some sense artificial: a continuous thing could be divided anywhere. Infinite divisibility is at the core of what it is to be continuous, and there is little disagreement about how to understand infinite divisibility. Unfortunately, understanding continuity is not so simple: infinite divisibility is necessary but insufficient for continuity. One might think, ‘Infinite division is a lot of divisions. Presuming one could actually divide infinitely, what would be left after such a process? Doesn’t Infinite Divisibility tell us enough about the continuum?’3 Infinite divisibility has mathematico-scientific consequences as well, independently of various purely conceptual concerns. Consider Euclid’s equilateral-triangle construction found in Book I, Proposition 1. Essential to this construction is the assumption that when two circles overlap one another, there are two points of intersection — one on top and bottom, as in Figure 1. 4 Fig. 1. View largeDownload slide Equilateral Triangle Construction from Euclid’s Elements. Fig. 1. View largeDownload slide Equilateral Triangle Construction from Euclid’s Elements. Suppose that the two circles have radius 2, and are centered at (1,0) and (–1,0) — points B and C in Figure 1. To complete the construction of the equilateral triangle, draw lines from C and B to one of the intersection points of the circles. Following Euclid, I chose the top one, labelled A. The problem is that the points of intersection would be at $$(\pm\sqrt{3}, 0)$$ — irrational values. It follows that these two circles do not intersect in the rational plane. Since the rational plane satisfies Infinite Divisibility, more is needed for continuity. At this point, the different conceptions of continuity radically diverge. 2. CONCEPTUAL TERRAIN Because the different conceptions diverge, allow me to map out some terrain. I will argue below that the five conceptions are divided over what mereological relation receives metaphysical priority in conceiving of the continuum: division or composition. 5 Or, equivalently, are the parts of the continuum fundamental or is the continuum as a whole fundamental? On the one hand, one way to conceive of the continuum is to begin with something continuous and try to analyze its parts qua parts. On this view, the continuum is itself fundamental; or, at least, it is more fundamental than any of its parts, let alone any points.6 Call this view top-down. On the other hand, one might view the parts of the continuum as fundamental. The continuum is then built up from its parts. Call this view bottom-up. Now for a closer look. 2.1. Top-Down Approaches Top-down approaches have their origins in antiquity with Aristotle.7 Although the Aristotelian conception of the continuum has significant appeal in its own right, top-down approaches emerge further upstream conceptually: the top-down orientation is visible within Euclidean geometry from which Aristotle takes his cues. Lines, planes, and solids are all treated as existing unto themselves to be divided by various constructions (for example, by a linear bisection) into parts.8 In that way, the top-down view is actually presupposed throughout the Elements.9 In addition to its relationship to geometry, the top-down approach undergirds a long-standing intuition that the continuum is unified through and through. Recall the discussion above regarding the distinction between discreteness and continuity. Discrete quantities exhibit distinctness and plurality; continuous quantities exhibit coherence and unity. The top-down approach provides a natural home for this unity intuition. The fundamentality of the continuum over its parts, let alone over points, delivers an image of the continuum primarily as a single unity and secondarily as composed of parts. Top-down approaches tend to view the parts of the continuum as being especially ‘sticky’: the parts are not merely arranged in a tidy order, but are in some sense ‘glued’ together. The continuum is thus sometimes described as being viscous.10 The way these intuitions get captured formally will be drawn out at length below. 2.2. Bottom-Up Approaches In contrast, bottom-up approaches — where the priority is on the relation of composition — are much newer historically. Seventeenth- and eighteenth-century mathematicians treated the continuum as composed of infinitesimally extended line segments; i.e., magnitudes $$m$$ so small that every integral multiple of $$m$$ is still strictly less than any chosen unit $$u$$ (for more detail, see § 4). Even still, when in philosophical mood, many of these mathematicians viewed infinitesimals as convenient fictions.11 Therefore, even though technical developments in modern mathematics suggested bottom-up thinking, it was not until well into the nineteenth century that a bottom-up picture emerged satisfying both mathematicians and philosophers alike. The first bottom-up conception of the continuum accepted by an overwhelming majority of mathematicians came from Richard Dedekind and Georg Cantor. Given the weight of tradition pushing heavily against this shift, what are some of the merits of the bottom-up approach? As I indicated, these merits are primarily bound up with technical developments internal to mathematics. The bottom-up approach fits naturally with an analysis that satisfies the highest standards of rigor. The fact that Dedekind-Cantor’s work developed in tandem with set theory and its analytic companion, point-set topology, further served to entrench the bottom-up approach. In other words, the bottom-up conception of the continuum was consonant with an overwhelming package of very fruitful mathematical tools. Compared with top-down approaches, bottom-up approaches prioritize mathematico-scientific values over traditional philosophical ones. 12 2.3. On Points Every view of the continuum has something to say about points and their relationship to the continuum. Historically, the most prominent question is whether that relationship is mereological; i.e., are points parts of the continuum? Call those conceptions of the continuum that answer in the affirmative, punctate. Within the set-theoretic venue of contemporary mathematics, non-null set-theoretic inclusion is the most natural way to interpret parthood in keeping with modern formal theories of mereology.13 Using this interpretation, the contemporary mathematical expressions of all but one of the conceptions are composed of points, independent of their historical provenance. It is for this reason that I take mereological priority — not punctation — to be more philosophically illuminating. No doubt, there are valuable, closely related, questions concerning points in the history of philosophy of mathematics. Nevertheless, this survey examines how these philosophical conceptions of the continuum have connected with and inspired recent mathematical work. 3. AN ORTHODOX CONCEPTION The best known conception of continuity comes from the mid- to late-nineteenth century. The combined work of Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, Richard Dedekind, and Georg Cantor is responsible for the way that the calculus and basic analysis is learned by all mathematics students even into the twenty-first century.14 Given its wide use and application, this conception may be properly called orthodox: (2a) Punctilious Points — An object $$X$$ is continuous if and only if $$X$$ is composed of a linear ordering of points and for any partition $$A,B$$ of $$X$$ such that all members of $$A$$ are less than all members of $$B$$, there is exactly one point $$c$$ that is either the greatest member of $$A$$ or the least of $$B$$.15 Below, I will further unravel some of the motivations behind this conception as well as examine why anyone has balked at this portrayal of the continuum. Finally, although Cantor developed an isomorphic form of the continuum, I focus on Dedekind. His philosophical commentary is more explicit and ultimately more transparent than Cantor’s. 3.1. Philosophical Considerations This conception plays a prominent role in part because it is among the earliest portrayals of continuity with no recourse to primitively geometric objects or spatial intuition.16 The mathematicians responsible for this view are not motivated as much by supplying an especially philosophically sensitive portrait of continuity. Rather, they want to provide a foundation for their activities that are consonant with their own standards of rigor. In previous times, explicit appeal to some kind of spatial intuition was acceptable, if not required. As mathematical standards shifted towards increased logical perspicuity, spatial intuition was relegated to pedagogical settings. The best way to understand this requirement is to return to the contrast between discreteness and continuity from above: discrete things are marked by their separateness. Dedekind’s aim here is to countenance the minimal resources to eliminate separateness. To sharpen things a bit, recall the example above from Euclid’s Elements (§ 1). When the two circles come together, there is supposed to be a point of intersection. The worry is, how can we be sure there is a point of intersection? Simply appealing to a sense that they should intersect will not suffice. The rational numbers are a well-defined class of values that fall along the continuum — what about them? Unfortunately, even the measure of sides and heights of mere triangles proliferate irrational values, as indicated by the example given above from the Elements. This is a weakness that cannot be ignored given that triangles are the simplest figures in plane geometry. In essence, Dedekind’s insight is to stop the gaps. Dedekind remarks on a similarly revealing example: The above comparison of the domain of rational numbers with a straight line has led to the recognition of the existence of gaps, of a certain incompleteness or discontinuity of the former, while we ascribe to the straight line completeness, absence of gaps, or continuity. [1872, p. 10, emphasis added] The idea of filling in the gaps left by the rationals was completely revolutionary.17 3.2. Reasons for Concern Other conceptions of continuity are significantly more stringent, anticipated by Dedekind himself. He writes, ‘[T]he majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed’ [1872, p. 11]. In this sense, Punctilious Points introduces just enough structure to serve certain primarily mathematical interests. This is exactly why people balk at his conception: it is at odds with the more traditional top-down approach. Its parts do not cohere enough — or at all. To be ‘viscous’ or ‘seamless’ to any extent seems to be undermined by the precise and analytical nature of contemporary formal mathematics. For many, the unity of the continuum is its premier feature. Indeed, to compose the continuous (a line) from the non-continuous (points on the line) is to subvert or entirely ignore this feature. This is to view the continuum as a hyper-plurality rather than a unity. Henri Poincaré affirms as much, only with approval: Of the celebrated formula, ‘the continuum is unity in multiplicity’, only the multiplicity remains, the unity has disappeared. The analysts are none the less right in defining their continuum as they do ... [1913, p. 43] For those familiar with the Dedekind picture, this is exactly how the continuum is viewed today: the continuum is best known for its uncountable cardinality. 4. INFINITESIMAL MAGNITUDES In Leibniz’s famous Theodicy, he writes: There are two famous labyrinths where our reason very often goes astray: one concerns the great question of the Free and the Necessary, above all in the production and origin of Evil; the other consists in the discussion of continuity and of the indivisibles which appear to be the elements thereof, and where the considerations of the infinite must enter in. [1952, p. 53] Given such a dramatic remark, one hopes that Leibniz18 would follow the Theodicy with an analogously systematic philosophical treatment of continuity. Unfortunately for us, he does not. As a decent substitute, he does supply some musings on his mathematical work, to be examined below. One finds him putting infinitesimals to great use in his own system of the calculus. In practice, Leibniz operates as if the following were true:19 (2b) Intransigent Infinitesimals — An object is continuous only if it is composed of infinitesimal magnitudes. 4.1. Philosophical Considerations A motivating picture for Leibniz is to think of a circle as an infinilateral regular polygon.20 Observe the sequence of regular polygons in Figure 2. On this way of thinking, the circle is composed of infinitesimally tiny lengths.21 In that way, this requirement is bottom-up — regardless of Leibniz’s official word on the matter.22 Fig. 2. View largeDownload slide Sequence of regular polygons converging to a circle. Fig. 2. View largeDownload slide Sequence of regular polygons converging to a circle. In order to license conceiving of circles and curves as composed of infinitesimals, Leibniz introduces his Law of Continuity, described in a variety of ways. In a letter to the adversarial Bernard Nieuwentijt, Leibniz writes: I take for granted the following postulate: In any supposed transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included. [1920, p. 147, emphasis original] To get a sense for how strong this statement is meant to be, consider how Leibniz puts his Law of Contintuity in another (more casual) way in a letter to Varignon, ‘the rules of the finite are found to succeed in the infinite’ [1956, p. 884]. One might put this as follows: if a sequence $$(s_n)$$ converges to some value $$S$$ and a property $$\varphi$$ is true of each $$s_i$$, then $$\varphi$$ is true of $$S$$. Allow me to apply this sequential interpretation to Leibniz’s circles-as-infinilateral-polygons. Each $$s_i$$ is a regular polygon with $$i \geq 3$$ sides and $$S$$ is a circle. Since geometrical properties of lines and polygons are fairly well behaved, they can be more easily studied than those of curves and circles. Take for example a regular pentagon ($$s_5$$). To derive its area one must simply find the area of one of the five identical isosceles triangles, the bases of which form the perimeter of the pentagon. Assuming one has the triangle’s base, one exploits the fact that the apex angle is $$2\pi/5$$ radians in order to find the height of the isosceles triangle. This relationship between the pentagon and these inner isosceles triangles ($$\varphi$$) is true of any regular polygon ($$s_i$$) with $$i \geq 3$$ sides. To get the area of the circle ($$S$$), extend this relationship ($$\varphi$$) to an $$\Omega$$ sided polygon, for some infinite number $$\Omega$$. The height of the isosceles triangle will be ‘identical’ to the radius, $$r$$, and the apex angle will be $$2\pi/\Omega$$. With some very simple algebraic manipulation, the proper result of $$\pi r^2$$ follows.23 Hence, extending this technique of finding the area of regular polygons ($$\varphi$$) succeeds in application to the circle ($$S$$). Let the reader note: this example is only meant to function as a rough-and-ready application of this line of thought. The utility of introducing infinitesimals extends far beyond such pedestrian exercises.24 4.2. Reasons for Concern There is a pair of concerns for Infinitesimals: one historical and one outstanding (of necessity). The now historical concern is the anxiety about, if not hostility towards, infinitesimals that stretches from Leibniz’s time right into the twentieth century. Bertrand Russell writes, And by [Leibniz’s] emphasis on the infinitesimal, he gave a wrong direction to speculation as to the Calculus, which misled all mathematicians before Weierstrass (with the exception, perhaps, of De Morgan), and all philosophers down to the present day. [1931, §303] As the reader will discover below, Abraham Robinson laid the charge of inconsistency to rest. The outstanding concern for Infinitesimals is its violation of the Archimedean property. Informally, the Archimedean property suggests that all magnitudes are comparable: if there is some difference between two numbers, the difference can be made up by integral multiples. In what follows, I will examine in turn both the consistency and (in)comparability of infinitesimals.25 4.2.1. Inconsistency: Ghosts of Departed Quantities One of the earliest and best-known attacks on infinitesimals comes from Bishop Berkeley’s The Analyst, with its rather forward subtitle, A Discourse Addressed to an Infidel Mathematician. Berkeley primarily complains that mathematicians treat infinitesimals equivocally — zero in one place and nonzero in another. He famously mocks this equivocation as follows: And what are these Fluxions? The Velocities of Intransigent Increments? And what are these same Intransigent Increments? They are neither finite Quanitites, nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities? [1992, §xxxv] This equivocation undermines the truth-preserving spirit of mathematical demonstration. The scientific fruit of the calculus is not enough: the ends might be good but they do not justify the means. Regardless of the Analyst’s strengths or weaknesses, practicing mathematicians largely ignored Berkeley. Mathematicians were aware of these kinds of issues, but the infinitesimal calculus was far too mathematically and scientifically fruitful for the Berkelean pestilence to ruin the harvest. The (temporary) eschewal of infinitesimals among mathematicians would not arrive for another century, following the epsilontic ($$\epsilon,\delta$$) revolution of the triumvirate of Cantor, Dedekind, and Weierstrass.26 Shortly thereafter, one finds Russell scornfully commenting in Berkelean spirit in his Principles of Mathematics: And we shall find that, by a strict adherence to the doctrine of limits, it is possible to dispense entirely with the infinitesimal, even in the definition of continuity and the foundations of the Calculus. [1931, §249] Hence infinitesimals as explaining continuity must be regarded as unnecessary, erroneous and self-contradictory. [ibid., § 324] When Russell wrote this, infinitesimals had made a post-triumvirate appearance in the work of Paul du Bois-Reymond and Otto Stolz, among others.27 These latter figures show that certain functions on real numbers behave like infinitesimals, thereby demonstrating that there is some sense to be made of infinitesimals.28 But, without a clear connection to standard results of analysis, it was perhaps too easy for naysayers to demur. In § 313 of Principles of Mathematics, Russell rather quickly dismisses this form of infinitesimal. In the mid-twentieth century, Abraham Robinson laid any doubt to rest with his nonstandard analysis (NSA). In [Robinson, 1974], he conservatively extends the first-order theories of real and complex analysis. This means that NSA does not prove anything about the real or complex numbers that could not already be proved by epsilontic methods. Therefore, Robinson’s infinitesimals were not just mathematically consistent (as in the case of, for example, du Bois-Reymond’s infinitesimals), but provably consistent with all analytical results. Hence, if there is a problem with infinitesimals, it cannot come from the charge of inconsistency. 4.2.2. Incomparability: Cholera Bacillus of Mathematics The second complaint against infinitesimals is that they are incomparable to ordinary real numbers, or rather, they violate the Archimedean property: for any values $$x$$ and $$y$$ where $$x<y$$, there is some positive integer $$n$$ such that $$nx >y$$. Infinitesimals are incomparable by their very nature. In some number systems, the infinitesimals are explicitly defined to be violations of the Archimedean property: $$\varepsilon$$ is an infinitesimal if and only if $$\varepsilon \ne 0$$ and for any positive integer $$n$$, $$\varepsilon< \frac{1}{n}$$; i.e., $$n \varepsilon <1$$. Famously, Cantor had a vitriolic antipathy for infinitesimals, calling them the ‘cholera bacillus of mathematics’.29 Cantor believed the Archimedean property is not merely an axiom to be held or not, but rather it is constitutive of the concept of linear magnitude.30 This is comparable to arguing that Euclid’s parallel postulate is constitutive of space: it follows from a representation of space more fundamental than an axiom. This passage sharpens one’s sense for Cantor’s concept of a linear magnitude: Thus the assumption we made [that an infinitesimal magnitude exists (for reductio)] contradicts the concept of a linear magnitude, which is of the sort that according to it each linear magnitude must be thought of as an integral part of another, in particular of finite linear magnitude. (Cantor, translated in [Ehrlich, 2006, p. 42], emphasis added) Here Cantor describes something suspiciously similar to the Archimedean property by requiring that ‘each linear magnitude ... be thought of as an integral part of another’. This requires that any linear magnitude, $$X$$, with length $$x$$, when summed with an identical magnitude yields another magnitude of length $$2x$$, and this process can be iterated indefinitely to $$3x$$, $$4x$$, $$5x$$, etc.. Furthermore, the above passage implies that such an $$X$$ be an integral part of a finite linear magnitude. In other words, for some positive integer $$n$$, $$n$$$$X$$s will compose a finite linear magnitude — impossible for an infinitesimal. Laying aside the details of the historical Cantor’s response to infinitesimals, this philosophical approach to magnitudes is largely at odds with a contemporary mathematical practice. To require linear magnitudes to fit this exact profile would be comparable to insisting that all spaces satisfy the parallel postulate. But, non-Euclidean geometry is among the greatest achievements of modern mathematics.31 Mathematicians are constantly searching to generalize and extrapolate the properties found in familiar mathematical objects. Ironically, Cantor’s own theory of infinite numbers is squarely at odds with long-held beliefs about number and size.32 Therefore, Cantor’s punctilious perspective on linear magnitudes is historically interesting, but not taken seriously by contemporary mathematicians. 5. SEMI-ARISTOTELIAN MATHEMATICS Aristotle’s view of the continuum can be characterized by roughly these features: (i) the continuum is not composed of points; (ii) points are dependent upon the continuum as mere locations or markers;33 (iii) infinity is potential.34 The discussion in this section will largely ignore (iii), given that the vast majority of the literature here surveyed has taken a completed infinity for granted.35 This conception may be captured as follows: (2c) Atomless Gunk — An object is continuous only if all of its parts have a proper part. It follows immediately that the continuum is top-down: there is no bottom! Aristotle writes: ... nothing that is continuous can be composed of indivisibles; e.g., a line cannot be composed of points, the line being continuous and the point being indivisible. (Physics, 231a23–25) Stated positively, this requirement suggests that, in dividing up the continuum, the parts will always be themselves continuous. On this view, are there no points at all? Points are available to play various roles as long as those roles are not mereological. Here is a summary of Aristotle’s view of points from Michael J. White: ... the existence of points presupposes the existence of magnitudes (or potentially divided ‘submagnitudes’) of which the points are the (potential or actual) limits ... Points have to be points of things that are not themselves points. [1992, pp. 16–17] This portrait not only underscores the top-down picture but also implies that points are ontologically dependent upon continuous things (referred to here as ‘magnitudes’). Orthodoxy has it the other way around: the continuum is dependent upon the points. In what follows, I will examine some of Aristotle’s thoughts on (i) and (ii) and their relationship to Atomless Gunk. Then I will outline some semi-Aristotelian mathematical incarnations. Let the reader note that not all of the mathematical literature surveyed in this section takes explicit inspiration from Aristotle, even if much of it does. Given the structural similarities between these theories, I hope the reader will indulge my referring to any or all of them as semi-Aristotelian. I will end this section with some concerns for Atomless Gunk. 5.1. Philosophical Considerations Allow me begin with a distinctive component of Aristotle’s own commentary, viz. continuity is treated as a dyadic relation and can be generalized to a multi-sorted one. So far, I have been primarily speaking of continuity and its cognates as a monadic predicate.36 In order to characterize continuity, Aristotle employs the following conceptual triad (in strict increasing order of logical strength): (A) Succession Some things are in succession if and only if they can be placed in some kind of order37 and nothing of the same kind is between them. In ordinary parlance, two things are in succession if one follows the other. (B) Contiguous Two things are contiguous if and only if they are touching. By touching, Aristotle has some kind of casual topological relation in mind; their extremities (boundaries) are ‘together’. For the moment, these notions need not be especially mathematically sophisticated. Given his examples, togetherness or touching can be sufficiently understood by togetherness or touching discernible with the naked eye. (C) Continuous Two things are continuous if and only if they are contiguous and form a unified whole. Aristotle distinguishes continuity and mere contiguity by the pairs’ extremities being ‘one’ rather than merely ‘together’. Consider the following concrete examples. Starting with succession and contiguity, dinner plates on a set table are in succession but are not contiguous — they are not touching. What about contiguity and continuity? Imagine a bottle filled with half water and half oil. For Aristotle, the oil and water in the bottle count as contiguous. They cannot be continuous, because they do not form a single uniform whole. If one were to replace the oil with all water, the pair consisting of the original water and the latter water would qualify as continuous.38 This example illustrates that continuity beyond contiguity is really a matter of the contiguous pair being constituted of the same ‘material’. In physical applications, the distinction is significant, whereas it is geometrically irrelevant since there is no such thing as geometric material. Indeed, it is most likely for this reason that there are no examples of Aristotle using ‘contiguous’ in reference to geometrical cases.39 Michael J. White provides the following link between the monadic concept of continuity and Aristotle’s dyadic relation. He calls it the non-supervenience of continuity: (N-SC) ‘Each partition of a continuous [monadic] magnitude into proper parts yields parts each of which is pairwise continuous [dyadic] with at least one other part.’ [1992, p. 29] What is the relationship between (N-SC) and Atomless Gunk? In order to see how Aristotle gets from (N-SC) to Gunk, I will present an argument Aristotle gives to that effect in Physics VI. Aristotle argues for Gunk by showing that the continuum is not composed of geometric points. In contemporary terminology, points are mereological simples, having no proper parts; in Aristotelean terms, points are indivisibles. For Aristotle’s argument to succeed, he assumes the following principle: ($$*$$) Mereological simples do not have extremities. Generally, mereological simples are themselves the extremities, but they do not have extremities. Now, we get the argument rolling: Suppose for reductio that a point, $$p$$, is part of the continuum. According to (N-SC), then $$p$$ is continuous with another part of the continuum, $$q$$. By definition, if $$p$$ and $$q$$ are continuous, they are contiguous; i.e., an extremity of $$p$$ and an extremity of $$q$$ are together. Contraposing ($$*$$), if $$p$$ has an extremity, then $$p$$ has a proper part. But, points have no proper parts, let alone extremities of any kind — contradiction (Physics, 231a24–28). Hence, points are not parts of the continuum, which is to affirm Atomless Gunk. Or as Aristotle epigrammatically puts it, ‘It is plain that everything continuous is divisible into divisibles that are always divisible ... ’ (Physics, 231b16). 5.2. Mathematical Expression Over the last century, several formal presentations of semi-Aristotelian continuity have been developed, many of which take cues from [Tarski, 1929], a work closely related to [Whitehead, 1929] and [Nicod, 1924]. However inspirational, Tarski signals that he has not provided a complete picture by referring to the work as a ‘sketch’ [1929, p. 24]. Others have followed in his wake, improving upon the details. Although there is a variety of approaches, all of them follow a basic recipe. Here are the ingredients: (A) Mereology A suitably strong mereological system is adopted to rule out tons of overly simplistic unintended models (modulo Skolemite considerations). Depending on the philosophical commitments of the author(s), there will be some variety in how arbitrary sums of parts are formed. This takes any of the familiar forms of employing or avoiding sets, ranging from outright quantification over sets (first- or second-order) to plural quantification to satisfaction of first-order sentential schemata. (B) Topology There must be some principles that require the parts to ‘bind together’ more robustly than a bare mereological sum. Of course, to use point-set topology would be question-begging; hence, familiar topological properties are introduced as primitive. Exactly which of these properties is taken as primitive depends on the author(s): some use topological connectivity, others use betweenness, others still construct points and then piggyback on point-set topological concepts. (C) Interpretability Once the conceptual pieces are in place, the author(s) show how (or assume that) one can recover everything afforded by the orthodox position. For some, they define points and prove that these constructed points behave just like the orthodox ones; for others, they prove that their theory of regions is isomorphic to the orthodox theory of points. Suffice it to say, there is no shortage of consistent modern semi-Aristotelean continua.40 5.3. Reasons for Concern The main reason anyone balks at this conception is due to its being a bit mathematically quixotic. The longstanding mathematical success of orthodoxy places a rather heavy burden on alternatives, especially if those alternatives involve costs for practice. This raises a larger question as to the goals of point-free programs. The elimination of points from geometry or analysis is akin in many ways to the nominalist effort to eliminate numbers from science. In the eyes of an observer, both of these programs raise analogous questions. The most significant one is, What relationship is the point-free or number-free theory meant to bear to the relevant science? As with nominalistic programs, different point-free programs offer different answers to those questions. In an effort to categorize these programs’ relationships to mathematical and scientific practice, I adopt an analogous distinction to one in the nominalism literature: (HP) Hermeneutical Pointlessness Mathematical and scientific practice may carry on as always; nevertheless, any statement making apparent reference to points can be given a point-free paraphrase. (RP) Revolutionary Pointlessness Mathematical and scientific practice should be altered in such a way as to end any reference to points.41. An example of someone who endorses (HP) is A.N. Whitehead: In modern language a point is often described as an ideal limit by indefinitely continuing the process of diminishing a volume (or area). Points as thus conceived are often called convenient fictions ... By calling the concept of points a convenient fiction, it must be meant that the concept does correspond to some important facts. [1929, pp. 137–138] Whitehead refers to points as convenient fictions, which has a very nominalist ring to it. Indeed, ‘convenient fiction’ is both a name and an appropriate description: points serve a practical purpose in representing what Aristotle called ‘limits’ or ‘extremities’ (convenient) but they do not really exist and theories about points are, strictly-speaking, false (fiction). Whitehead is content to allow science to carry on as always, so long as in our serious moments, we refer only to regions instead. The major problem for (HP) ironically comes from the point-free programs’ successes. To have mutual interpretability between pointy and point-free theories is important for the program to be taken seriously by practitioners, especially for the (HP) variety. However, once the work is done, the question is raised: what is the difference? By comparison, the algebraic relation of partial order can be defined using a strict or non-strict ordering: the former denies the property of reflexivity and the latter embraces it. Since these axioms of partial order are mutually interpretable, it is really a matter of preference or ease of use in some particular venue of application. Likewise, it is hard to see what genuine difference there is between treating the points as primitive or the regions. The way to success for a point-free theory is to offer something that goes beyond the work done by the pointy theory.42 Until that happens, debating whether space is composed of points or regions is comparable to debating whether a partial order’s relation is reflexive or not.43 To give a more everyday example, the mutual interpretability result suggests that the debate about the composition of space is like two people arguing about whether a dime has a portrait of Roosevelt or an image of a torch: they are two sides of the same coin. What about the Revolutionary Pointlessness? (RP) has been defended in some form by Frank Arntzenius. I say ‘some form’, because he is quite cautious in his claims, especially in later writings.44 That being said, of those authors I have read on these matters, he is one of very few who has seriously explored the question as to whether the physical sciences could be done using point-free mathematics.45 In [Arntzenius, 2003], he holds a more aggressive stance on the eschewal of points. There he argues that formal features of separable Hilbert spaces encourage us to think of these spaces as point-free. He also attempts to defuse the common claim that quantum field theory requires points and argues that with minor changes to the formalism, quantum field theory actually supports Atomless Gunk. In more recent work, Arntzenius’s tone is noticeably less sanguine: ... there appears to be no single devastating argument that space and time (or matter) have to be gunky. Nonetheless, it remains of interest to examine the possibility of doing physics in gunky space and time in more detail. [Arntzenius, 2008, p. 234] He has cooled towards Gunk but has not completely abandoned it as unworthy of investigation. In the same article, Artzenius betrays a rather Quinean outlook on the relationship between ontology and scientific practice: Electric fields, mass densities, gravitational potentials, etc. ... are standardly represented as functions from points in space and time to point values. This practice would seem to make no sense if time and space did not have points as parts. [2008, p. 225, ellipsis and italic emphasis original, bold emphasis added] The emphasized statement above suggests that Arntzenius expects scientific practice obediently to reflect the metaphysical structure of the world. In Quine’s words, our ‘canonical scheme’ — the regimented language of science — should be ‘limning the true and ultimate structure of reality’ [Quine, 1960, p. 221]. Arntzenius implies that pointy space-time may well be thrust upon us by scientific practice (even as he pursues the possibility of a point-free physics).46 This perspective is very much like Quine’s famous indispensability argument, viz., that scientific practice thrusts mathematical objects upon us (but if we could, we would do without them). Given the similarity between Arntzenius’s perspective on points and Quine’s perspective on numbers, I claim that Burgess’s and Rosen’s response to Quinean indispensability sheds light on (RP) independently of exactly where Arntzenius falls on the matter. What do Burgess and Rosen say? The substance of their complaint against Quine is that the indispensability argument is ‘a major concession to nominalism’ [Burgess and Rosen, 1997, p. 64]. If one can successfully demonstrate that physical sciences could be done without numbers, then the nominalist wins. Elsewhere, Burgess alone writes, ‘I maintain, however, that science at present is done with numbers, and that there is no scientific reason why in future science should be done without them [Burgess, 1983, p. 95]. The Quinean stance that scientific practice must sound a clear note concerning its objects — or else — is sustainable only under a highly idealized, irenic image of scientific activity.47 In general, scientists and mathematicians adopt the objects most natural to their immediate purposes with only periodic interest in how these systems of objects interrelate. The exception to this rule is when scientists (or mathematicians) discover scientific (or mathematical) reasons why cross-disciplinary efforts will bear fruit. This indicates that Quine’s taste for desert landscapes is in many ways at odds with the rather liberal ontologizing that marks the sciences.48 Rarely does one scientific discipline go as far as posit the existence of entities that are intrinsically incompatible with the entities of another discipline. In those cases where a conflict does arise, scientists commit themselves to both kinds of entities and carry on until something emerges to settle the conflict, in the form of either a unified theory or a decisive piece of empirical evidence. In the spirit of Burgess and Rosen, I take issue with Arntzenius’s claim that the practice of representing physical features using functions on points ‘would seem to make no sense if time and space did not have points as parts’ [Arntzenius, 2003]. Perhaps points are not well-suited to the quantum mechanical, infinite-dimensional apparatus of Hilbert spaces — no problem. The fact that one branch of science is better off without points should not impugn another branch’s liberal use of them. Forget about points for a moment. Quantum mechanics is also better off without making reference to living organisms or currency markets. Likewise, zoology and economics both get by quite well without a single mention of a positron. As far as I am concerned, there are particles, organisms, markets, and much more. So, is space composed of points or regions? Pick a side: heads or tails. 6. INDECOMPOSABILITY AND EXCLUDED MIDDLE The Intuitionistic Indecomposables requirement takes the top-down approach the most seriously. Hermann Weyl [1925] favorably reports of Brouwer’s continuum, ‘A continuum cannot be put together out of parts’ (p. 135, emphasis original). The parts are dependent on the whole so much that they cannot stand alone: (2d) Intuitionistic Indecomposables — An object is continuous only if it is not identical to a sum of any of its disjoint proper parts. This conception has its mathematical origins in the intuitionism(s) of L.E.J. Brouwer and Hermann Weyl, though I will also examine Smooth Infinitesimal Analysis, developed by Anders Kock and F.W. Lawvere. 6.1. Philosophical Considerations Indecomposables is the strictest conception of the continuum. Much of the motivation for this strictness emerges from Kantian features found in both Weyl and Brouwer’s thinking. In this survey, I will primarily focus on Weyl’s philosophical writings, because (i) they are far more lucid than Brouwer’s, and (ii) they have borne more recent mathematical fruit beyond the original seeds of intuitionism.49 Lastly, for the reader’s ease, I will write intuition with the small caps to indicate the distinctively Kantian meaning, leaving all other uses to fit more contemporary usage. The only exception to this rule will occur in quotations. Weyl and Brouwer attempt to employ a Kantian conception of mathematical thought broadly, and to capture an intuitive continuum in particular. Weyl praises Brouwer’s mathematical program with the following Kantian sentiments: With Brouwer, mathematics gains the highest intuitive clarity; his doctrine is idealism in mathematics thought to an end. [Weyl, 1925, p. 136] ... It is greatly beneficial that Brouwer has strengthened again the sense in mathematics for the intuitively given. (ibid., p. 141) By referring to the likes of the intuitively given, Weyl is no doubt invoking Kant, specifically his epistemology of mathematics and his insistence on the place of intuition therein. Without wading too deeply into the Kantian quagmire, this at least involves seeing mathematics as epistemically grounded in the forms of intuition — the inner subjective features of the mind that structure and govern all possible experience, but are not themselves subject to the vagaries of actual experience — as opposed to those broadly logical notions at the epistemic foundation of Dedekind’s and Cantor’s work.50 From this return to an intuition-sensitive epistemology of mathematics, a very robust picture of the continuum emerges. Weyl writes: Within a continuum, one can very well generate subcontinua by introducing boundaries; yet it is irrational to claim that the total continuum is made up of the boundaries and subcontinua. The point is, a genuine continuum is something connected in itself, and it cannot be divided into separate fragments; this conflicts with its nature. [Weyl, 1921, p. 111, bold emphasis added] The above quotation is somewhat interpretatively delicate, given that the two statements seem to suggest opposite things about the continuum. The first suggests one can divide the continuum; the second suggests that one cannot divide the continuum. The first passage is almost a rephrasing of the statement of Intuitionistic Indecomposables. Regarding the second passage, the appropriate reading of ‘cannot be divided into separate fragments’, is not that there are no parts of the continuum at all — a tempting way to read things; rather, one should read it that the nature of any part of the continuum is fundamentally as a part. In other words, parts of the continuum do not have any identity or ontological status beyond their place within the continuum: they are nothing more than parts of the continuum.51 In any case, Weyl clearly defends a picture of the continuum where the parts of the continuum take an ontological back seat while the continuum itself is up front in the driver’s position. Along with Kantian inspiration, Weyl and Brouwer are unified in their anxiety about Punctilious Points. To see Weyl’s own assessment of the orthodox conception, consider the following picturesque description: To represent the continuous connection of the points, traditional analysis, given its shattering of the continuum into isolated points, had to have recourse to the concept of a neighborhood. [Weyl, 1921, p. 115, bold emphasis added] From a historical perspective, Weyl is correct. The concept of a neighborhood (roughly, a set of points around a point) gives rise to a whole topology on the continuum, $${\mathbb{R}}$$. As Weyl suggests, the introduction of a topology on $${\mathbb{R}}$$ is necessary since the orthodox conception of the continuum — called ‘traditional’ here by Weyl — itself is very minimal. The orthodox conception has to impose additional structure on the continuum in order to get its parts to ‘stick together’. Weyl believes that the continuum should already be together. Weyl’s image of shattered orthodox continuum echoes Poincarés words above: ‘only the multiplicity remains, the unity has disappeared’ [1913, p. 43].52 6.2. Mathematical Expression Here I will briefly analyze two separate mathematical expressions of Intuitionistic Indecomposables: Intuitionistic Analysis (int) and Smooth Infinitesimal Analysis (sia).53 6.2.1. Intuitionistic Analysis Intuitionistic Analysis emerges from the larger intuitionistic program that takes numbers to be mental constructions of a certain kind. The inspiration is predictably Kantian: if number is rooted in the pure intuition of time, then it follows likewise that number ‘in itself, outside the subject, is nothing’ (A35/B51). In more contemporary terminology, numbers are mind-dependent entities. From this mind-dependence, a number of restrictions on the construction of the intuitionistic continuum, $${\mathfrak{R}}$$, emerge: (A) Potential Infinity Since human beings have finite minds, and numbers are mental constructions, it follows that all infinities in int are only potentially infinite; that is, the intuitionistic counterparts of classically infinitary objects are at best indefinitely finite. There is, of course, some measure of idealization, but the intuitionist jettisons the possibility of a finite mind apprehending a completed infinity.54 (B) Rejection of Excluded Middle Since the mathematical universe is essentially mind-dependent, mathematical statements are not made true by corresponding to some diaphanous, platonic realm; statements only gain truth values by being proved or refuted in human history. Therefore, statements of the form $$(\varphi \vee \neg \varphi)$$ may not be asserted without either a proof or refutation of $$\varphi$$: a disjunction may be affirmed only with a proof of at least one of its disjuncts. (C) Predicativity Mathematical objects gain their ontological purchase through the activity of defining. A definition is impredicative if the definiendum is among the objects cited in the definiens. A familiar impredicative definition is the least upper bound of some set $$S$$: the standard definition requires mentioning the class of upper bounds of $$S$$, of which the least is one. Such impredicative definitions are forbidden since — according to the intuitionist — they seem to assume a pre-existing universe to be examined, as opposed to seeing the defining as itself a creative act. To the intuitionist, impredicativity is like ontological boot-strapping. Along with these restrictions, a striking novelty is introduced: the concept of a choice sequence.55 Choice sequences are like ordinary sequences but adjusted for the requirements (A) and (B) above. For (A), at any moment in time only a finite initial segment of a sequence can be given. For (B), a sequence’s values are under-determined by that initial segment, though it is permitted that some (not all!) behave according to some explicitly articulated rule.56 Between the restrictions above and the (provable) peculiarities of choice sequences, it follows that all functions defined on $${\mathfrak{R}}$$ are uniformly continuous. The indecomposability of $${\mathfrak{R}}$$ follows immediately.57 6.2.2. Smooth Infinitesimal Analysis The inspiration for sia is radically different from that for int. Rather than being driven by a specific ontology and epistemology of mathematics (like intuitionism), sia emerges from work in algebraic geometry using category-theoretic techniques.58 The basic object of sia is the smooth real line, $$\mathscr{R}$$. In sia, the ‘viscosity’ of $${\mathscr{R}}$$ is due to special properties of its infinitesimals, the nilsquares, $$\Delta$$.59 A nilsquare is any value $$\varepsilon$$ such that $$\varepsilon^2=0$$. What makes $${\mathscr{R}}$$ so viscous is the Kock-Lawvere Property:   \begin{equation} (\forall f: \Delta \rightarrow \mathscr{R})(\exists ! m \in \mathscr{R})~(\forall \varepsilon \in \Delta)~~~ f(\varepsilon)=f(0)+\varepsilon \cdot m. \end{equation} (KLP) The idea behind (KLP) is curves are linear on the nilsquares. Notice that (KLP) is the natural representation for differentiation within sia where $$m=f'(0)$$, in conventional notation.60 Since (KLP) essentially requires that every function be differentiable, it follows that every function is continuous and therefore $${\mathscr{R}}$$ is indecomposable. Like int, sia requires an intuitionistic logical consequence relation. How is that? This is due to a tension between (KLP) and that $${\mathscr{R}}$$ is a field.61 The field axioms guarantee that every nilsquare is indistinguishable from zero (i.e., for all $$\varepsilon \in \Delta, \neg \neg \varepsilon = 0$$). To see this, suppose for contradiction that there is some nonzero nilsquare $$\varepsilon$$. If $$\varepsilon \ne 0$$, then it has a multiplicative inverse, $$1/\varepsilon$$. Now, since $$\varepsilon^2=0$$, it follows that   \begin{equation} 0=\varepsilon^2 \cdot \frac{1}{\varepsilon} = \varepsilon\cdot \Big(\varepsilon\cdot \frac{1}{\varepsilon}\Big) =\varepsilon\cdot 1=\varepsilon. \end{equation} (†) Tracing the equality through (†), it follows that $$\varepsilon=0$$ — contradiction. Therefore, there are no nonzero nilsquares. In classical logic, one infers immediately that all nilsquares are zero ($$\Delta=\{0\}$$); however, this contradicts (KLP). How exactly? For simplicity, set $$f(x)=x+1$$. Substituting into (KLP),   \begin{equation} (\exists ! m \in \mathscr{R})~(\forall \varepsilon \in \Delta) \underbrace{f(\varepsilon)}_{=\varepsilon+1}=\overbrace{f(0)}^{=0+1}+\ \varepsilon\cdot m. \end{equation} (‡) However, by the assumption $$\Delta =\{0\}$$, (‡) above is equivalent to $$1=1+0\cdot m$$; i.e., $$0=0\cdot m$$. The problem is that $$0=0 \cdot x$$ for all$$x$$, not just this $$m$$. So, $$m$$ is not unique — contradiction. Hence, $$\Delta \ne \{0\}$$. Backing up to get the big picture, all this implies that unless the logic of sia is weakened so that $$\Delta=\{0\}$$ does not follow from (†), sia would be inconsistent. Like similar features of $${\mathfrak{R}}$$, these subtle logical adjustments serve to bind tightly the parts of $${\mathscr{R}}$$.62 6.3. Reasons for Concern The most striking problem for this requirement is that all extant theories satisfying this requirement permit a logic no stronger than intuitionistic logic. Here is the requirement again: (2d) Intuitionistic Indecomposables — An object is continuous only if it is not identical to a sum of any of its disjoint proper parts. To demonstrate this, assume the Law of Excluded Middle (lem) for reductio. Denote the continuum with $$\mathcal{R}$$ and the positive values with $$\mathcal{R}^+$$. Now, let $$x \in \mathcal{R}$$. By lem, $$x >0$$ or $$x \not > 0$$. Therefore, either $$x \in \mathcal{R}^+$$ or $$x \in \mathcal{R}\setminus \mathcal{R}^+$$. In other words,   \[ x \in (\mathcal{R}\setminus \mathcal{R}^+) \cup \mathcal{R}^+, \textrm{which implies } \mathcal{R} \subseteq (\mathcal{R}\setminus \mathcal{R}^+) \cup \mathcal{R}^+. \] But obviously both of these are subclasses of $$\mathcal{R}$$ and so,   \begin{equation} \mathcal{R}=(\mathcal{R}\setminus \mathcal{R}^+ )\cup \mathcal{R}^+. \end{equation} (**) The standard way of understanding parts in a set-theoretic universe is by treating set inclusion as the parthood relation and arbitrary unions as sums.63 Therefore, (**) says that the continuum is composed of the disjoint positive and non-positive parts. But by Intuitionistic Indecomposables, $$\mathcal{R}$$ is not identical to a sum of any of its disjoint proper parts — contradiction.64 What this shows is that in order to satisfy Intuitionistic Indecomposables, one must either give up this natural interpretation of parthood within the set-theoretic setting, or one must abandon lem.65 The extant theories satisfying Intuitionistic Indecomposables all choose the latter option. In each case, there are other reasons to abandon lem that are closer to the conceptual center of the theories. So, this is not the only reason why lem is abandoned. In any event, the consequences of adopting such strong requirements on the continuum are often savored — rather than choked down — by their adherents. Now, beyond the necessitation of intuitionistic logic, are there more direct concerns with those continua satisfying Indecomposables? Consider the following. A central result in classical analysis is the intermediate value theorem (IVT), which ensures that there is a locus of intersection for curves in space (i.e., continuous functions). According to (IVT), if a function, $$f$$, is continuous on interval $$[a, b]$$ and $$f(x_1)>0$$ and $$f(x_2)<0$$ for some $$x_1, x_2 \in [a, b]$$, then there is some $$x_3 \in (a, b)$$ such that $$f(x_3)=0$$. This result is not only a part of our basic spatial intuition but has very deep roots in the science of mathematics. Recall the example from Euclid’s Elements above (§ 1). The existence of the point of intersection of the two circles is presupposed by Euclidean geometry: when two continuous things (whether lines or curves) cross one another, there is a point of intersection. Does IVT hold for either of $${\mathfrak{R}}$$ or $${\mathscr{R}}$$? Strictly speaking, the answer is no. Analysts working with $${\mathfrak{R}}$$ and $${\mathscr{R}}$$ make the following apologia for such disappointing results. They admit that the classical IVT is provably false,66 but urge us to accept a weakened version of IVT. For $${\mathfrak{R}}$$, instead of asserting that there is some $$x_3$$ (as above) such that $$f(x_3)=0$$, one may prove instead that $$f$$ can get arbitrarily small.67 For $${\mathscr{R}}$$, the only extant intermediate-value principle available places additional requirements on $$f$$ to deliver the result.68 In the case of $${\mathfrak{R}}$$, these apologists point out, the alternative versions of IVT are classically equivalent to the classical IVT. This equivalence is often cited as a reason not to abandon int or sia. I see two problems with this apologia. First, the classical IVT provides a point of intersection between continuous lines, whereas IVT’s counterparts cannot guarantee this perfectly natural geometric intuition (on pain of contradiction). Such a difference implies that these systems’ logical tweaks purchase philosophical intuitions (the unity of the continuum) at the expense of geometric control (a guaranteed locus of intersecting continua). Again, this tension runs right along the border between philosophical ideals and scientific practicality. Second, just because one can supply a classical equivalent does not mean that a classicist cannot see the intensional differences; after all, a classicist qua mathematical logician can see just as well that for each of int and sia, classical IVT is false and its counterpart is true. Any metalogical results concerning the equivalence of IVT and some counterpart are not the exclusive province of the intuitionist mathematician employing them in his defense. 7. PEIRCE AND THE SUPERMULTITUDINOUS Charles Sanders Peirce’s conception of the continuum represents another attempt to reject the bottom-up view of Punctilious Points.69 Peirce initially favored the Dedekind-Cantor conception of the continuum as indicated by his early article [1892]. Eventually, he rejected it as a mere ‘pseudo-continuum’ [Peirce, 2010, p. 218]. Peirce was restless in his meditations about continuity, progressing from more scientifically viable conceptions to more philosophically sensitive ones based on the phenomenology of time. Since Peirce’s continuum was constantly changing, it may come with no surprise that, at one time or another, he endorsed all of the other four requirements above.70 Therefore, I have isolated one requirement that sets him apart: (2e) Prodigious Possibilities — An object $$X$$ is continuous only if for any cardinal number $$\kappa$$ and for any two distinct points on $$X$$, there are at least $$\kappa$$ parts between those points. In what follows, I will examine some of Peirce’s own remarks on the continuum followed by discussion of its contemporary mathematical expression in the surreal numbers. Finally, I will count the cost of holding to such a baroque conception of the continuum. 7.1. Philosophical Considerations Peirce’s conception of the continuum is driven by the notion that a continuum manifests all possible magnitudes, roughly expressed, in the requirement Prodigious Possibilities. Given Peirce’s negativity about Points, the reader might infer (correctly) that Peirce adopts a top-down prioritization. Here is an explicit definition: [A] continuum is whatever has the following properties: 1st, it is a whole composed of parts ... 2nd, these parts form a series ... 3rd, taking any multitude whatever, a collection of those parts can be found whose multitude is greater than the given multitude. [Peirce, 2010, p. 163] The first two properties are simply to indicate that a continuum forms a linear array, as opposed to some disorganized jumble of parts.71 However, the third property gestures at Prodigious Possibilities, where Peirce uses the expression multitude to mean the more conventional cardinal number. This third property demonstrates Peirce’s top-down orientation: the idea is meant to mimic Atomless Gunk while remaining sensitive to the Cantorian infinite. The idea is that one cannot get to the bottom of the continuum. Peirce’s innovation is to see the endless heights of the set-theoretic hierarchy within the depths of the continuum. The following argument for Prodigious Possibilities, found in Peirce’s voluminous Nachlass, captures his top-down spirit.72 Begin with a line segment $$\ell$$ in the Euclidean plane: a simple geometric line unburdened with presuppositions from orthodoxy. Now, divide $$\ell$$ once into identical parts: there are two line segments $$\ell_1$$ and $$\ell_2$$. If instead one divides $$\ell$$twice into identical parts, there will be three line segments $$\ell_1, \ell_2$$ and $$\ell_3$$. One could continue this process indefinitely. Generalizing, if one divides $$\ell$$, $$\kappa$$ times, there will be $$\kappa+1$$ parts, $$\ell_1, ..., \ell_{\kappa+1}$$. For Peirce, the line is divisible as many times as possible, where possibility extends as far as all cardinal numbers. The extent of possibility for Peirce captures a central aspect of his contempt for Dedekind-Cantor’s ‘pseudo-continuum’ — it fails to plumb the depths of possibility fully.73 Therefore, for any cardinal number $$\kappa$$, the line is divisible $$\kappa$$ times and therefore there are at least $$\kappa+1$$ parts. It follows from this that the parts of the continuum form a proper class, i.e., it is so large that it cannot be a member of any other class. To offer a sense of scale, proper classes may be placed into one-one correspondence with the entire set-theoretic universe.74 7.2. Mathematical Expression Following insights from [Ehrlich, 2010], I claim that the field75 of surreal numbers, No, best express Peirce’s conception.76 The surreals resurrect infinitesimals comparable to the hyperreal numbers that undergird NSA (§ 4.2.1). As one might expect given the demands of Prodigious Possibilities, there is a proper-class-size worth of values between any two points. In the surreal jungle, the familiar 0, 5, and $$\pi$$ make appearances; more surprisingly, one also finds the following more exotic flora: $$\omega-1, \frac{1}{\sqrt{\omega}}$$ and $$\omega^{\frac{1}{\omega}}$$, where $$\omega$$ is the first limit ordinal. In fact, the surreals generalize the real numbers and transfinite ordinals, On. This embedding of the ordinals in the surreals is an easy indicator that No is a proper class. How do the surreals satisfy Prodigious Possibilities? Since the surreals include the class of ordinals, one may draw on its multiplicative inverses to define the following:   \[ S_\alpha=\bigg\{ x \in \it{No}: \frac{1}{\alpha+2} \leq x \leq \frac{1}{~\alpha+1~}\bigg\}, \alpha \in \it{On}. \] Each $$S_\alpha$$ is a part of the surreal continuum.77 Finally, since the cardinals are defined in terms of ordinals, and the $$S_\alpha$$ range over all ordinals, it follows that for any cardinal $$\kappa$$, there are at least $$\kappa$$$$S_\alpha$$-parts. Now, the $$S_\alpha$$s cover the interval (0,1]. To generalize, a simple linear transformation of the $$S_\alpha$$s using arbitary points $$a,b \in {\it{No}}$$ will suffice to demonstrate that No satisfies Prodigious Possibilities. 7.3. Reasons for Concern What difficulties are there for Peirce’s continuum as expressed by the surreals? As with the other top-down conceptions of the continuum, there is a concern that the surreals are not well-suited to mathematical applications. Of course, unlike the others that lack certain critical theoretical features, the surreals have an embarrassment of riches. Therefore, I will argue below that the surreals might function in various applications but not necessarily in a way that could supersede anything in orthodoxy; indeed, no extant efforts even attempt wholesale replacement.78 How have the surreals been used? First, the surreals were inspired by John Conway’s examination of moves in the ancient Chinese game, Go. The book, On Numbers and Games is so-called because the surreals form a special subclass of a larger class of games. Alongside applications in game theory, some applied mathematicians have taken an interest in the surreal numbers because the surreals provide a singular venue for discussing a variety of mathematically significant concepts. The surreals are not so arcane as to be ignored by practicing mathematicians. The question becomes, how useful are the surreals? In spite of interest in the surreals and some possible applications, there have been long-standing obstacles to developing a full analysis of the surreals. It should be admitted that the surreals can function as a model of NSA (§ 4.2.1), but then the surreals do not offer anything special not already available in the basic models of NSA, the hyperreals. When it comes to an analysis over No itself, the largest outstanding problem is in developing a surreal integral. The most substantial contribution to this problem is found in the joint work of Ovidiu Costin, Philip Ehrlich, and Harvey Friedman.79 On the positive end, they successfully employ the surreals in very current research in asymptotic analysis, a discipline central to applications; nevertheless, work remains to be done, as explicitly acknowledged by the authors.80 8. CONCLUSION I have examined five major requirements one might place on the continuum: when expressed in contemporary mathematics, each of them yields a radically different image of the nature of continuity. I have also attempted to offer a pair of parallel insights that I believe help focus the discussion. First, a continuum’s punctation (or lack thereof) is traditionally the focus of discussions of the continuum; however, in applying modern mereological formalism to the conceptions’ contemporary mathematical expressions, one discovers that only one fails to be punctate. To capture more closely what distinguishes these conceptions from one another, I have introduced the alternative distinction of bottom-up versus top-down. Second, this distinction runs right in line with tension between the conceptions: the bottom-up conceptions favor practical, mathematico-scientific needs while top-down conceptions favor more intuitive, philosophical values. 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For the sake of this survey, extended will apply to any geometric body (however construed) that is not a single point. This implies that extended applies to lines, planes, and surfaces as well as mereological sums of points scattered through space. Something is bounded if and only if all of it can be contained within a sphere of the appropriate dimension. Hence, the rational line ($${\mathbb{Q}}$$) counts as infinitely divisible. 2For each example of discreteness that he gives, he concludes, ‘[they] are separate’ and again, ‘they are always separate.’ (4b27, 4b31) 3For Aristotle, Infinite Divisibility suffices — at least practically — for continuity. This is partly due to two distinctly Aristotelian assumptions: (i) points cannot be parts of a continuum, discussed at length in § 5; (ii) infinity is only potential (discussed in greater detail in § 6.2.1 (A)). These assumptions rule out the rational line as an immediate absurdity, not worthy of even asking if it satisfies Infinite Divisibility. The concept of a rational number line is as manifestly absurd to Aristotelian ears as the square circle is to ours — a line by its very nature cannot be made of a completed infinity of points. 4The original way this example gets used is to show that Euclid’s own axioms are not sufficient for establishing the existence of these two points. For discussion on this and many other fascinating issues, see for example [Shabel, 2004] or [Friedman, 1992], especially ch. 2. 5Throughout the survey, I freely use mereological expressions, such as part, sum, compose, and so forth. These concepts are given formal treatments in both [Casati and Varzi, 1999] and [Simons, 1987]. Straightforwardly combining this work with insights from [Lewis, 1991], the mathematics behind all conceptions except Intuitionistic Indecomposables satisfy General Extensional Merelogy’s axioms. Indecomposables falls far short, failing to satisfy even Weak Supplementation. 6Discussion of metaphysical priority and related notions can be found, for example, in [Schaffer, 2009]. Formal treatments are offered by Fine [2012] and Rosen [2010]. I would like to remain neutral about the exact nature of metaphysical priority, given that this is a subject of current discussion and research. For the sake of this survey, I take fundamentality, metaphysical priority, and metaphysical dependence to be those notions at the center of the familiar Euthyphro dilemma. In particular, that dilemma is often constructed as follows: (1) Is God more fundamental (metaphysically prior to) moral properties? If so, then God capriciously determines what is right and wrong by fiat. (2) Are moral properties more fundamental than (metaphysically prior to) God? If so, then God is subordinate to something. I take the notion of metaphysical dependence to be the dual of metaphysical priority; i.e., if $$A$$ is prior to $$B$$, then $$B$$ depends upon $$A$$. To illustrate in a more nearby discussion, platonists and intuitionists take the following respective stances: (3) Human mathematical activity depends upon the nature of mathematical universe; (4) The nature of mathematical universe depends upon human mathematical activity. The foregoing is a somewhat coarse presentation of those views, but I hope it serves in providing a sense of the relevant concepts. 7Perhaps the top-down language is terminologically infelicitous with respect to the historical Aristotle. For him, continua are bounded above and below, unlike the unbounded continua of contemporary mathematics. Hence, Aristotle’s continuum is top-down in virtue of its being bottomless (see §5). 8Euclid uses the language of part-whole, including using even μέρη — the word from which we get ‘mereology’ — in the original. See for example, Book V, Proposition 15. 9In Euclidean geometry, points and lines play very distinct roles. The notion that points compose a line is neither explicit nor implicit in the Elements. 10To my knowledge, this use of ‘viscous’ comes from Carl Posy. 11For example, Leibniz is known for viewing infinitesimals as fictions. In spite of Leibniz’s otherwise liberal usage in his mathematical work, his preferred metaphysical view of the continuum was very much top-down. In a paper called ‘Infinite numbers’, he writes, ‘In the continuum, the whole is prior to its parts; the absolute is prior to the limited; and so is the unbounded prior to that which has a bound’ [2001, p. 97]. Leibniz is slightly equivocal, which makes him a rather mysterious figure. His elusive fictionalism allows him to maintain these two voices. For further discussion on Leibniz’s fictionalism, see [Levey, 2008]. 12I suspect that such a tension between mathematical fruitfulness and philosophical fidelity is what pushed philosophically sensitive mathematicians, like Leibniz, towards viewing infinitesimal magnitudes as merely fruitful but unacceptable for the ‘official business’ of philosophy. 13See note 5. Following [Lewis, 1991] and explicitly invoking mereological language, a continuum $$\mathfrak{C}$$ is punctate, if for each member $$m$$ of the linearly ordered set $$\mathfrak{C}$$, the singleton $$\{m\}$$ (the point) is a part of $$\mathfrak{C}$$ (the line). One might object to this interpretation as follows: $$\{m\}$$ is actually just the singleton of the point $$m$$, not the point itself; so this is not an appropriate interpretation of what it would mean for the line to be composed of points. While it is true that this is indeed the singleton of what is traditionally considered the point, what counts as a point is largely determined by other structural factors in play. The number $$m$$ is the point when looking simply at the order structure, $$(\mathfrak{C}, \leq)$$ as an arithmetical representation of the line. However, under Lewis’s interpretation, singletons are mereologically simple: they have no proper parts, which is exactly how Euclid defines point. That is to say, the structure under consideration where $$\{m\}$$ is a point is not $$(\mathfrak{C}, \leq)$$ but rather $$(\mathfrak{C}, \subseteq)$$. In this latter structure, $$m$$ itself is neither a point, nor anything at all — the structure does not have the resources to say anything at all about it. 14See [Rudin, 1976, Appendix to Chapter 1]. Rudin’s textbook is still a benchmark in analysis courses for advanced undergraduate and beginning graduate students. 15This definition prohibits higher dimensions. One could view this as the base case for an inductive definition. The inductive case is as follows: an object is continuous$$_{n+1}$$ if and only if any projection into $$n$$-dimensions is continuous$$_n$$. 16The intuition discussed here is very much like what Kant means when he refers to space as a form of pure intuition. 17One might wonder, how can one be sure Dedekind got all the gaps? The gaps to which Dedekind is referring correspond to any place on the continuum to which a rational Cauchy sequence would converge — if it could — but cannot. (Roughly, a sequence is Cauchy if arbitary differences between sequence members converge to zero.) The number systems developed by Dedekind and Cantor ensure that all Cauchy sequences converge — including sequences with irrational members; i.e., they are Cauchy complete. 18I focus on Leibniz because of the seminal role that he played in the infinitesimal calculus. 19Notes 11 and 12 above briefly address some complications related to differing tones in Leibniz’s voice on infinitesimals. 20Leibniz is not the first person to think of a circle this way. Several sources place the origination of this notion with Democritus. Many others in the immediate historical vicinity of Leibniz treated circles and other similar curves — e.g., cycloids — as infinilateral polygons, including Descartes, Kepler, and the Bernoullis. For more details, see [Bos, 1974; Edwards, 1979; Katz and Sherry, 2013]. 21The Kock-Lawvere Property, discussed in § 6.2.2, likewise encourages viewing curves as composed of infinitesimal line segments. The theory of infinitesimals discussed there is far more restrictive than this one, as the reader will discover below. 22See note 11. 23Explicitly, the $$\Omega$$-gon has area: $$A_\Omega=\Omega (bh/2).$$ Using the apex angle, we exploit the fact that the isosceles triangle splits into two right triangles. Therefore, since $$h=r$$, $$b/2=r\tan(\pi/\Omega).$$ Substituting, it follows that, $$A_\Omega= \Omega r^2 \tan(\pi/\Omega)=\pi r^2,$$ since $$\tan \theta=\theta$$ for infinitesimal $$\theta$$. 24Leibniz’s Law of Continuity is meant to apply even to examples that one would not expect. There are several other cases beyond the infinilateral polygon to which he applies his Law. For example, he uses his Law to analyze the behavior of parallel lines in such a way that they behave like other pairs of lines. In so doing, he treats all lines as intersecting, where parallel lines intersect ‘at infinity’, and the angle between them is infinitesimal. Such examples might make us balk, but this strange Law paid substantial mathematical dividends. 25As far as I can tell, the term incomparables comes from [White, 1999]. 26For lively discussion, see [Kitcher, 1983] and [Katz and Sherry, 2013]. 27Ehrlich [2006] discusses the work of du Bois-Reymond, Stolz, and other pre-Robinsonian non-Archimedean systems from this period, a small sample of which includes [Hilbert, 1899], [Levi-Civita, 1892/3]. An English summary of du Bois-Reymond’s work can be found in [Hardy, 1910]. 28More precisely, when comparing continuous real-valued functions at infinity, one finds unmistakably infinitesimal behavior. For example, define $$f(x)=1/x$$ and $$g(x)=1$$. It follows that for each positive integer $$n$$, $$n\cdot f \prec g$$, where $$x \prec y$$ is defined by,   \[ \lim_{t\rightarrow \infty} \frac{x(t)}{y(t)} =0, \textrm{ for }x, y \in {\mathbb{R}}^{\mathbb{R}}. \] In other words, the rate of growth of $$f$$ is infinitely smaller than the rate of growth for $$g$$. 29Cantor puts it thus in a letter to Vivanti found in [Meschkowski, 1965], quoted in both [Ehrlich, 2006], and Chapter 5 of [Grattan-Guiness, 1980]. 30A full treatment is provided in § 8 of [Ehrlich, 2006]. 31A delightful historical irony: not only is the parallel postulate unnecessary, it is physically false. 32For discussion, see [Mancosu, 2009]. 33Aristotle writes, ‘That which is divisible in two dimensions is a plane, that which is divisible in one a line, that which is no way divisible in quantity is a point or a unit, — that which has not position a unit, that which has position a point.’ (Metaphysics$$\Delta$$, 1016b28–30, emphasis added) 34Potential infinity is discussed in greater detail in § 6.2.1(A). 35Linnebo et al. [2016] provide a full portrait of (i)–(iii) using modal logic to capture potential infinity. 36This difference is not meant to be especially deep. It is more a matter of presentation. To speak of continuity as being one way or the other is simply a matter of whether we choose to speak of the sum of some things as being continuous or of the things together being continuous. Furthermore, it is worth noting that Aristotle speaks both ways. In the Categories, Aristotle says, ‘A line, on the other hand, is a continuous quantity’ (Categories, 4b36–5a1) In any case, in those sections relevant to the present discussion, Aristotle primarily uses the dyadic conception of continuous. 37Although Aristotle does not spell this out explicitly, one may assume that the order relation of Succession is at least partial: irreflexive, anti-symmetric, and transitive. 38I borrowed this example with some modifications from [White, 1992]. 39White notes, ‘There is for Aristotle, I suspect, no topological distinction between parts that are contiguous and parts that are continuous. Rather continuity pertains to what is homeomerous, while contiguity pertains to parts which are spatially joined but essentially different’ [1992, p. 27, emphasis original]. 40For more details, see [Roeper, 2006] and [Roeper, 1997], as well as [Hellman and Shapiro, 2013]. For related discussion, see [Casati and Varzi, 1999, esp. chs 3–5]. 41To my knowledge, this distinction is first found in [Burgess, 1983], and echoed in [Burgess and Rosen, 1997] The way I use these expressions is not directly analogous to how they are used by Burgess and Rosen, even if there are similarities. 42Johnstone [1983] articulates some distinct advantages for a pointless topology. Nevertheless, these are given as advantages that supplement rather than supplant point-set topology. Pointless topology is one more valuable branch of mathematics alongside point-set topology. 43Another useful comparison comes from determining whether the set-theoretic counterpart of zero is a member of the counterpart of two: on one representation it is, on another it is not. In neither case is the theory of arithmetic affected by this difference: zero is still less than two. Cf. [Benacerraf, 1965]. 44In [Arntzenius, 2012], he is the most tentative in his conclusions. His chapter Pointlessness concludes ambivalently with the resigned, ‘Oh dear ...’ (p. 152, ellipsis original). 45Hellman and Shapiro explore this question briefly in their monograph [forthcoming]. Compared to Arntzenius, their commentary approaches the question primarily from the mathematical side: what impact would a point-free theory have on mathematical tools employed in mathematical applications, viz. measure theory and differential geometry? Because of their roughly Carnapian approach to the metaphysical questions, they do not commit to anything as robust as (HP) or (RP). 46Hellman and Shapiro are openly suspicious of this kind of metaphysical inquiry. One might ask, what are they doing that has not been already done by mathematicians? My personal sense for their works’ contribution is that they use concepts more familiar to the average philosopher: first-order logic, plurals, etc. The primitive notions they use are straightforwardly geometric and do not require any further mathematical background. There are other more subtle advantages but I will leave the reader to examine those for herself. 47Penelope Maddy, from studying some episodes in the history of science, retreated from mathematical realism, exemplified in [Maddy, 1990], to a position of ontological ambivalence, exemplified in [Maddy, 1997] and [Maddy, 2007]. I am not convinced by the conclusions she has drawn, but agree with her assessment that Quine’s science is too simplistic. 48Burgess and Rosen [1997, §III.C.1] argue against Quinean philosophers of science who prioritize Occam’s Razor and the maxim of theoretical simplicity over all other theoretical virtues, especially familiarity, fecundity, and perspicuity. Notably, Arntzenius explicitly expresses a preference for these same maxims favored by Quine in his [2012]. 49The philosophical literature on Weyl’s own mathematical work is somewhat limited. Weyl develops a predicative analysis (see § 6.2.1 (C) below) but without any strictly intuitionistic features [1918]. Weyl eventually abandoned that particular program for a more intuitionistic one, remarking in later editions that he did not have time to make the appropriate adjustments in line with his newer views (see the Preface to the 1932 edition). Much of his own philosophical commentary is more closely linked to this later intuitionism. The limited secondary literature on his earlier work is not helped by the fact that [Weyl, 1918] was written using idiosyncratic notation, rendering it nearly impervious to an uninitiated reader. Feferman [1998] offers an excellent discussion of this text. Feferman also takes care to translate Weyl’s notation into contemporary mathematical notation. On Weyl’s intuitionism, see [Bell, 2000] and [van Dalen, 1995]. 50The exact nature of these Kantian invocations is noteworthy. What makes these mathematicians’ interest in Kant so significant is their eagerness to adopt his much broader philosophical outlook, rather than specific features of his philosophy of mathematics. Brouwer in particular acknowledges some of the historical Kant’s difficulties following the development of non-Euclidean geometries: However weak the position of intuitionism seemed to be after this period of mathematical development, it has recovered by abandoning Kant’s apriority of space but adhering the more resolutely to the apriority of time. [Brouwer, 1913, p. 57] 51Thanks to Lisa Shabel for helpful discussion of this passage. 52It is a peculiar irony that the intuitionists are drawn to Kant’s perspective on the pure intuition of time which Kant uses in his philosophy of arithmetic; i.e., not his theory of geometry, the native soil of the continuum. Even as the intuitionists decry the orthodox shattering of the continuum, it is historically striking that they likewise pursue an arithmetic development of the continuum. 53There are a number of systems of analysis in the neighborhood (no pun intended) of these two. There is Errett Bishop [1967]’s constructive analysis, developed into the present by Douglas Bridges [1985] among others. There is also Russian constructivism developed by A.A. Markov Jr (to be distinguished from his also very famous mathematician father of the exact same name [Kushner, 2006]). Bishop’s system is the weakest. Each of classical analysis, Russian constructive analysis and Brouwer’s Intuitionistic Analysis extend Bishop’s system in mutually inconsistent ways; however, sia is not an extension of Bishop’s system as they are mutually inconsistent. (See [Bridges, 1994] and [Bell, 2001].) It is worth special note that Bridges and Richman [1987, p. 120] claim that one can neither prove nor refute the continuity of all functions on Bishop’s continuum. This is significant because indecomposability is constructively equivalent to the claim that all functions are continuous. For this reason one might say Bishop’s continuum is non-decomposable: neither decomposable nor indecomposable. 54Extended discussion can be found in § 3.1 of [Dummett, 2000]. 55Troelstra argues in both [1982] and [1983] that Brouwer’s perspective on the continuum shifted away from a top-down continuum (holistic, in Troelstra’s terms) to a bottom-up continuum (analytic) as Brouwer’s emphasis on choice sequences increased. Independent of the historical Brouwer, I strain to supply a philosophical interpretation of the mathematical property of indecomposability (satisfied by $${\mathfrak{R}}$$; see note 62) as anything other than top-down: the parts are so tightly bound together as to be incapable of separation. This tension strikes me as more a larger puzzle within Brouwerian scholarship and less one that abrogates my claim that Brouwer’s continuum is top-down. 56The reader will find detailed discussion of choice sequences in Chs 3 and 7 of [Dummett, 2000] and in Chs 4 and 12 of [Troelstra and van Dalen, 1988]. 57This is due to the result that if $$f$$ is continuous, then any function of the following form is constant:   \[ f(x)=\left\{ \begin{array}{ll} 0 & \textrm{if}\,\,x\in U \\ 1 & \textrm{if}\,\,x \in V,\end{array}\right. \] where $$U,V$$ partition the continuum. It follows that exactly one of $$U$$ or $$V$$ is empty. 58A brief but engaging history can be found in [Moerdijk and Reyes, 1991]. See also [Bell, 2008]. 59For a brief gloss on viscosity and its cognates, see § 2.1 and footnote 10. 60This can be generalized by simple linear transformations to apply to any point, not just zero. Also, the reason (KLP) is written as an equation in product form is that $$0 \in \Delta$$ and so one cannot simply divide by all of the $$\varepsilon \in \Delta$$ as in the usual expression of the derivative as a quotient. That is, since $$0 \in \Delta$$ and $$\varepsilon$$ ranges over $$\Delta$$, the following expression is bogus in sia:   \begin{align} m=\frac{f(\varepsilon)-f(0)}{\varepsilon} \end{align} 61A field is a class closed under addition and multiplication: both operations are associative and commutative and multiplication distributes over addition; there are unique additive and multiplicative identities; finally, there are additive inverses for all values and multiplicative inverses for all values except for the additive identity. 62One might think that an indecomposability result would be an all-or-nothing matter. It turns out that $${\mathfrak{R}}$$ is far more ‘sticky’ or ‘viscous’ than $${\mathscr{R}}$$. Observe these differences in the following table, with exemplars of each category. Varieties of Decomposability between $${\mathscr{R}}$$ and $${\mathfrak{R}}$$. $$\mathcal{R}$$ and $$\mathcal{Q}$$ generically refer to either of $${\mathscr{R}}$$ or $${\mathfrak{R}}$$ and their rational subsets respectively. This suggests that $${\mathfrak{R}}$$ is so sticky that even when a hole is cut out — indeed, infinitely many holes — the remaining parts cannot be broken apart. Further details can be found in [Bell, 2001] and [van Dalen, 1997]. 63See note 5. 64A very similar argument is given in [Posy, 2005]. 65Strictly speaking, there is another option: deny the existence of points. This option is discussed at length above (§ 5). Carl Posy argues that a pointy continuum that satisfies Intuitionistic Indecomposables is a significant philosophical contribution from the intuitionists. He writes, ‘Brouwer has shown us that we can have a viscous [indecomposable] continuum and make it out of points as well. This is no windmill-tilting: this is a profound mathematical insight.’ [Posy, 2005, p. 347] 66For int, a counterexample is found at [Troelstra and van Dalen, 1988, Vol. I, p. 292]. For sia, a counterexample is supplied at [Moerdijk and Reyes, 1991, pp. 317–318]. 67A proof of this is found in [Bishop, 1967, p. 40]. It is also in [Troelstra and van Dalen, 1988, Vol. 1, p. 293]. Another intermediate-value principle is supplied in the latter citation in the same section but where greater restrictions are placed on the function. 68Details on IVT in sia can be found in [Moerdijk and Reyes, 1991, pp. 315–318]. Thanks to John L. Bell for helpful correspondence on this topic. 69Historically speaking, Peirce tended to interact more with Cantor’s provably equivalent mathematical expression of Punctilious Points. 70For discussion of Peirce’s progression, see [Moore, 2007] and [Moore, 2015]. 71Recall that I claim that all conceptions satisfy Infinite Divisibility. In other words, I do not expect the requirements given by (2a)–(2e) to reflect every aspect of a given conception. 72This argument can be found in [Peirce, 1976, Vol. 3], with the editorial title, ‘[Multitude and continuity]’. 73At this point, readers of Peirce enter into very dark waters. The continuum for Peirce not only expresses all numerical possibilities, but he claims that all metaphysical possibilities are structured like a continuum: The continuum is concrete, developed possibility. The whole universe of true and real possibilities forms a continuum, upon which this Universe of Actual Existence is ... a discontinuous mark — like a line figure drawn on the area of a blackboard. [Peirce, 1976, Vol. 4, p. 345] 74This can be demonstrated in von Neumann-Bernays-Gödel (NBG) set theory. In VIII §3, Bernays [1968] shows that one can place $$On$$ into one-one correspondence with the universal class $$V$$. With this result, one can show that any proper class $$C$$ can be placed into one-one correspondence with $$V$$: it is a rather straightforward exercise to biject $$On$$ onto $$C$$ using transfinite induction. The principle that a proper class is equinumerous with the universal class is elsewhere referred to as Limitation of Size. 75See note 61. 76This requires a couple of brief but necessary qualifications: (1) Ehrlich actually excludes the infinite values to stay harmonious with the historical Peirce. In my estimation, all of $$No$$ suffices for a survey’s level of detail. (2) Wayne C. Myrvold constructs a Peircean continuum in [Myrvold, 1995]. Since it is primarily for a relative consistency result, it lacks some of the richness displayed by Ehrlich’s clipped surreals; most notably, it is not absolutely dense. For further detail, see n. 34 of [Ehrlich, 2010]. 77Parthood here is interpreted as in [Lewis, 1991]. 78Cf. note 42. 79See [Costin et al., 2015]. Some earlier contributions to surreal integration include some direct but limited treatment in [Fornasiero, 2004] and some brief discussion in [Conway, 2001]. 80Costin et al. make use of the theory of transseries, an accessible introduction to which can be found in the playful survey by Edgar [2009/10]. © The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
TI - Labyrinth of Continua
JO - Philosophia Mathematica
DO - 10.1093/philmat/nkx018
DA - 2018-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/labyrinth-of-continua-g8q63Z5GD0
SP - 1
EP - 39
VL - 26
IS - 1
DP - DeepDyve
ER -