Continuous Frustration: C.S. Peirce’s Mathematical Conception of Continuity

 

Introduction

In 1898, Charles Sanders Peirce gave a series of Cambridge lectures on the relationship between mathematics and ontology.  Peirce insisted that this relationship—between a specific system of human reasoning and being itself—had to be one of intimacy.  Peirce’s insistence, however, was particularly fervent.  He repeatedly quoted his father, Benjamin Peirce, as saying that mathematics provided “a science of drawing conclusions from arbitrary hypotheses,” and therefore stood as the purest form of inquiry into the nature of being.  In the Cambridge lectures, Peirce attempts to explain the concept of continuity in terms of the more primitive notions of Kanticity and Aristotelicity.  This paper exposes the way in which this project fails, and fails for very specific reasons. This argument, however, is only a preparatory move in the project at hand.  In time, Peirce comes to realize that a purely set theoretic approach could not fully describe the character of true continua.  He does not, however, jettison his hope that true continuity could be explored mathematically.  In the Minute Logic of 1902, Peirce once again emphasizes the purity of mathematics, but also categorizes the pure hypotheses from which mathematics must follow.  These realms of hypotheses can be organized according to the character of the collections or sets in question.  Thus, hypotheses are organized according to whether they have to do with finite collections, infinite collections, or true continua.  

            The primary aim of this paper is to underscore the importance of these three categories.  We argue that the distinctions that Peirce makes in 1902 form the basis of a mathematical architectonic that provides a solution to the frustration he faced in 1898.  Most of the secondary literature has overlooked this vital moment in the Minute Logic.  Here, Peirce recognizes that the most of his difficulties stem from conflating these three modes of mathematical hypotheses and neglecting the necessarily phenomenological aspects of true continuity.  In 1902, he begins to make a turn in his thinking on continuity, a turn that Potter and Shields believe that Peirce makes in 1908 in his “post-Cantorian period.”[1]  Whether the Minute Logic provides a positive mathematical conception of true continuity remains unclear.  What is clear, however, is that Peirce recognizes the deficiencies in his 1898 lectures and provides an alternative model.  In conclusion, the paper argues that Peirce’s third conception of mathematical hypotheses, that is the realm of true continua, can be investigated only in understanding the role of mathematics and phenomenology in grounding Peirce’s conception of the normative sciences.  Peirce addresses the interesting relationship between mathematics and phenomenology in his work on diagrammatic and topical geometry.  Here, in the twilight of his career, Peirce suggests that a type of intuition or insight—rather than proof—furnishes the foundation for the mathematics of true continua.

A word also needs to be expressed on the various attempts that have been made in the secondary literature to mathematically unravel Peirce’s conception of continuity.  Most of the secondary scholarship has attempted to explain Peircean continuity in terms of other concepts in mathematics.  For example, Wayne Myrvold discusses Peirce’s concept of the continium in terms of set theory and Cantor’s paradox.[2]  Hillary Putnam and Kenneth Ketner creatively interpret Peirce’s continuum as a prototype of contemporary non-standard analyisis.[3]  Timothy Herron reworks some of Peirce’s formulations of continuity into modern mathematical notation and then compares and evaluates those formulations with respect to modern theory about infinitesimals.[4]  Jean Louis Hudry examines Peircean continuity as pure (non-metrical) geometry and concludes that it cannot be a foundational mathematical concept due to its lack of application in other mathematical fields.[5]  In contrast to these other writers, we do not attempt to explain continuity in terms of other concepts in mathematics, but instead aim to expose Peirce’s own challenges in his mathematical negotiation with continuity.  These challenges encourage a turn in Peirce’s thinking that is central to our paper.  This turn occurs when Peirce envisions the hypotheses of true continua as being distinct from other areas of discrete mathematics.  This paper attempts to elucidate these unique hypotheses. 

First, it is worthwhile, and indeed necessary, to briefly outline the principle of continuity that emerges in the course of his 1898 Cambridge Lectures.  While Peirce flirts with the principle in earlier years, the Cambridge lectures provide an overview of Peirce’s sustained investigation on the subject and a point of departure for this paper.

 

I.  Thinking Through Continuity: Early Attempts

Peirce’s approach in his 1898 lectures was determined, at least in part, by his reading of Kant and Aristotle.  These figures present two valuable accounts of continuity, accounts that Peirce both appropriates and critiques.  Throughout the 1890s, Peirce was engaged with George Cantor’s ideas, which also proved formative in his thinking on continuity.

In the “Law of Mind” (1892), Peirce roughly defines “Kanticity” as the property of infinite divisibility.  This property implies both geometric and arithmetic divisibility, summarized in turn: that a line consists of infinitely divisible parts, and that a line is defined by an infinite divisibility of parts.  The Kantian continuum, construed as a geometrical line, is constituted by its points.  Peirce acknowledges that Kant provides at least one quality of true continua; as Kelly Parker notes, “To say that a continuum is infinitely divisible, is to say that it has no ultimate parts.”[6]

Aristotle develops a notion of continuity which is, according to Jean Louis Hudry, a “compound of arithmetical convergence and the geometric sharing of a common limit.”[7]  This rendering of continua relies upon a property which states that, for every convergent series of points on a line segment, the limit of that series is contained in the line segment.  By employing the notion of limit, Aristotle maintains that points do not belong to a line, or constitute a line, but lie on the line.  Points designate particular ways in which the line can be demarcated or cut.  Aristotle envisions an infinite series of points as an infinite series of possible divisions of a line.  He suggests that this series has an infinite convergence point and that this point cannot be reached in a finite series of steps.  Peirce writes, “Aristotelicity may be roughly stated thus: a continuum contains the endpoint belonging to every endless series it contains…every continuum contains its limit.”[8]

Peirce came to see that Kanticity was not sufficient, that “gaps” in the continua can still exist in Kant’s conception of continuity.  For example, the union of the disjoint line segments (-∞, a) and [b, ∞), a < b, is dense and, thus, continuous according to Kanticity.  Denseness does not necessitate connectivity.  As Parker notes, Kant fails to account for the connection-relation between the parts of the continuum.[9]  In lieu of this shortcoming, Peirce employs Aristotelicity, whose immediate consequence is that gaps in the continuum cannot occur, for the following reason.  There is a sequence of real numbers on the line segment (-∞, a) whose limit is a, for example the sequence sn = a – 1/n, which gives us limn→∞(sn) = a.  But a is not contained in the line segment (-∞, a).  Since Aristotelicity guarantees that every limit is contained in the line, the union of the line segments (-∞, a) and [b, ∞) is not continuous.  Note that Peirce’s critique of Kant and appropriation of Aristotle produces a model of the continuum that is consistent with Cantor's.[10]  He carries this model into the Cambridge lectures of 1898.

Peirce opens the lectures reflecting on his indebtedness to Aristotle, stating, “I am Aristotelian and a scientific man.”[11]  His indebtedness becomes clear when he begins to outline his definition of continuity.  Peirce asks his audience to envision a geometric line that has been cut at a marked point and then that the subsequent line segments are pulled apart.  He suggests that this cut defines two distinct internal endpoints.  Peirce then asks a revealing question:  “Where is that marked point now?”  Peirce’s demonstration suggests that the single point, according to the geometrical or phenomenological intuition of the observer, becomes two separate distinct endpoints.  He then goes on by asking the observer to imagine those endpoints being “joined together” by pushing the line segments back together.  He notes that, “if the junction ceased to have any distinguishing character, that is any discontinuity [any mark], there would not be any distinct point there.”[12] 

This demonstration appears to refute Peirce’s earlier implicit acceptance of Cantor’s identification of the geometrical line with the set of real numbers.  In modern mathematics, when a cut is made in the real number line, it is called a Dedekind cut.  This cut divides the real number line into two segments, like Peirce’s construction, but only one of those segments has a proper endpoint.  We may divide the real number line at the number a and obtain two segments, (-∞, a) and [a, ∞).  This is not a violation of Aristotelicity.  As above, there is a sequence sn on (-∞, a) whose limit is a.  The interval (-∞, a) does not contain this limit, but the union of the intervals (-∞, a) and [a, ∞) does contain it, since a is contained in the line segment [a, ∞).  Thus, every limit of every convergent sequence of real numbers is contained in the union of (-∞, a) and [a, ∞), so the union is continuous.  Since a is any arbitrary real number, Dedekind cuts preserve continuity.  What troubles Peirce, however, is the idea that the interval (-∞, a) corresponds to a geometrical line segment obtained by cutting a line, because the interval (-∞, a) is, by itself, not continuous, since (-∞, a) does not contain the limit a of sn.  Thus, Peirce’s thought experiment indicates an even more powerful move towards an Aristotelian conception of the geometric line, one that must work with geometrical or phenomenological intuition as much as with Kanticity.

Peirce recognizes that any endless series cannot be completely counted.  Nevertheless, “[w]e have a conception of the entire collection of whole numbers.  It is a possible collection indeterminate yet determinable”[13]  In contrast to the concrete activity of counting, Peirce is allowing the mathematician to conceive of that activity as if it were completed.  This move seems to echo Aristotle’s conception of continuity insofar as distinct points are markers for possible division. Similarly, a line, conceived as a collection of points, where each point is an endpoint for an endless series of other points, can be conceived hypothetically as a possible collection of divisions or discontinuities.

Peirce has shown that Cantor’s conception of a line does not accord with geometrical intuition.  In true pragmatic fashion, Peirce balks at the notion that mathematical and metaphysical claims can be out of kilter with phenomenological insight.  This point will prove formative in Peirce’s later development of the mathematical architectonic in which he suggests that the possibility of true continua can be grasped, and then only in part by a type of phenomenological investigation.  In drawing Cantor’s assumptions into question, Peirce asks a concerning question: What is the cardinality of the continuum if not that of the real numbers?  By way of his line-cutting demonstration, Peirce seems to have shown that the continuum has a higher cardinality than that of the real numbers.  It would seem that one point could give rise to two.  Peirce claims that, from a single endpoint, “points might fly off in multitude and order like all the irrational quantities from 0 to 1.”[14]  At any one point in the continuum, there is a vast potential for further differentiation.[15]  The possible always outstrips the actual.  A mathematician, therefore, can conceive of the line on the continuum in abstraction only as the totality of possible differentiation.  Peirce states:

            … (A) continuum is a collection of so vast a multitude that in the whole

universe of possibility there is no room for them to retain their distinct

identities; but they become welded into one another.  Thus the continuum

is all that is possible.[16]

 

In 1898, Peirce begins to recognize this “welding,” this connectivity, as the distinctive character of true continua.  Despite this fact, he is still wrestling with—or trying to maintain—his idea expressed in earlier years that there are “discrete multitudes out of which continuous multiplicity can be compounded.”[17]

The tension between determinability and indeterminacy of sets that characterizes Peirce’s early work reaches a certain apotheosis in his final Cambridge lecture.  Once again, Peirce insists that he still want to recognize distinct individuals in the continuous multitude.  It seems that Peirce is still profoundly ambivalent: “Individuals are determinable in every multitude.  That is, they are determinable as distinct.  But there cannot be a distinctive quality for each individual.”[18]  He goes on to state that there must be “relations” that distinguish any individual from another.  At this point, it still unclear what Peirce means by “relations.”  His ambivalence is confusing at best, incomprehensible at worst.  It is, however, this ambivalence that his 1902 work seeks to explain in its development of a mathematical architectonic.    

 

II. Dividing Mathematics – Unifying the Continuum

When Peirce began the 1902 Minute Logic, he intended to construct a twenty-three-chapter work on the foundations of mathematics and logic.  It was, more generally, to lay out the groundwork of right reasoning in all fields of inquiry, from aesthetics to ethics to mathematics.  What was actually constructed was literally minute in comparison: three and a half unpublished chapters in which Peirce undertakes a classification of the sciences and mathematics.  Peirce begins this work by stating that “mathematical truths are derived from observation of creation of our own visual imagination.”[19]  Our visual imagination, embedded in phenomenological experience, supplies at least the rudiments of these hypotheses.  The tension between indeterminacy and mathematical determinacy is replaced by the challenges of formalizing a branch of reasoning that can take account of true continua in phenomenological character.  In 1902, Peirce’s understanding of true continua begins to reflect this turn in thinking.  He begins to unify the continuum by dividing mathematics, by making space for phenomenology within a particular vein of mathematics.

            In the first section of the Minute Logic, he elaborates on his father’s belief that mathematical truths are drawn from arbitrary hypotheses, stating that “the hypotheses of mathematics relate to systems which are either finite collections, infinite collections, or true continua; and the modes of reasoning about these three are quite distinct.”[20]  Peirce begins to flesh out these three modes of reasoning, but only completes the section devoted to finite or denumerably infinite collections.

            In dealing with finite or denumerably infinite collections, Peirce suggests that one is essentially dealing with a mathematics of logic—“dichotomic” and “trichotomic logic.”[21]  A finite collection can be defined as a denumerable collection of objects and that any of these objects under consideration possess fixed and determinable identities.  Since these objects are denumerable, there is a general method with which one could exhaustively determine the set.  Consider two mathematical examples.[22]  Suppose we take the set of positive integers smaller than five, i.e., {1,2,3,4}, and the property of evenness.  There is a method whereby we may separate this set into two sets, one whose elements truthfully bear that property and one whose elements do not.  One initiates this method by selecting the smallest element and moving to the largest, and in this case obtains the sets {2,4} and {1,3}.  Now suppose we are dealing with the denumerably infinite set of rational numbers, and suppose the property in question is expressed in the predicate, “strictly less than ½.”  There is a method, namely Cantor’s diagonalization method, by which we may enumerate this set and, in so doing, determine for each element whether or not it possesses the property in question.  The force of these examples is to show that this category of mathematics deals with a certain kind of collection, and this kind of collection allows for a certain kind of reasoning.  This reasoning allows for a truth value to be “attached” to each and every member of a collection, relative to some property.[23]

            Nondenumerably infinite collections—that is, collections with aleph-one or larger cardinality—cannot be reasoned about in the same way as finite or denumerably infinite collections.  Abnumerable multitudes retain distinct individuals; for example, each and every real number can be uniquely defined as an infinite sequence of rational numbers.  One may take any single real number and determine whether or not it has a certain property, just as in finite and denumerably infinite collections.  However, by virtue of their nondenumerability, there is no method of enumerating every single real number and, for each, determining whether or not it has a property.  Nonetheless, there is a way of reasoning about these collections that does not require enumeration but which relies upon general characteristics of a collection.  For example, one may seperate the real numbers into two sets, those that truthfully and those that falsely possess the property expressed in the predicate, “the square root of -1.”  By virtue of general characteristics of real numbers, we obtain an empty set (no real number is the square root of -1) and the set of all reals.  This way of reasoning is also present in finite and denumerably infinite collections.

            These two modes of mathematical reasoning—reasoning by individual enumeration and reasoning by general characteristics—are associated with particular collections of distinct individuals.  Distinctness remains the guiding principle in each case; truth values can be “attached” to members in either mode, relative to some property.  Peirce, however, remains unsatisfied with the two modes.  As Hausman observes, Peirce refuses to define a continuum “so it permits the correlation of its elements with a series of discrete, quantitative units.”[24]  Instead, he proposes a third conception: the mathematical hypothesis of true continuity.  The type of reasoning proper to this branch of mathematics, in contrast with the first two types, will not regard continuity as a gathering of discrete individuals.  In the Minute Logic, Peirce provides only the most vague suggestion as to the method by which true continua can be investigated, commenting that true continua corresponds to the investigation of “topical geometry.”  At the end of his 1898 lectures, Peirce addresses topical geometry but often employs diagrammatic geometry in order to elucidate his thinking.  Scholars such as Ketner, Hudry, and Putnam have shown that modern topology is inconsistent with Peirce’s notion of true continuity.  These observations, while accurate, should not distract one from Peirce’s comment on the relation between true continua and topical geometry.  Peirce had something else in mind when he referred to topical and diagrammatic geometry as a way of investigating continuity.

            To frame Peirce’s understanding of geometry, it is necessary to return to the concept of Peirce’s architectonic, to the categories by which he hopes to organize the normative sciences.  He concludes in 1903 that these normative sciences rest on both mathematics and phenomenology.  A year earlier, he states quite ambiguously that mathematics rests on observation, while at the same time, insisting that “phenomenology is dependent on the conditional or hypothetical science of Pure Mathematics.”[25]  When it comes to an investigation of true continua, and for that matter when it comes to the issue of topical geometry, there must be an overlap or a type of reciprocity between these seemingly distinct fields of thought.  For Peirce, the pairing of mathematical hypothesis and phenomenological observation provides access to the nature of true continuity, an access that could not be achieved by either discipline in isolation or in their traditional renderings.

 

The Mathematics of True Continua:

Phenomenology, Intuitive Abduction, and Diagrammatic Thought

 

            At first glance phenomenology and mathematics seem worlds apart.  In the twilight of his career, however, Peirce assumes the project of unpacking creative, non-deductive aspects of mathematical reasoning.  In effect, he aims to highlight a certain reciprocity between phenomenology—what he called phaneroscopy—and the forms of hypothesis that could ground a mathematics of continuity.  Peirce suggests that not only phenomenology but also aesthetics might shed light on the branch of mathematics that deals with true continua.[26]  As Kathleen Hull observes, Peirce’s later work on non-deductive reasoning has often been misconstrued as non-mathematical.  Following Douglas Anderson’s lead, however, she argues that these non-deductive ways of reasoning, exemplified in reasoning by diagram, ought to be considered mathematical.  Indeed, diagrammatic thought is the only way that mathematics can account for Peirce’s mature understanding of true continuity.  Hull describes this mature view aptly when she writes that “continua are collections of possibilities, and these possibilities are not fully determinate objects and lack distinct individuality.”[27]  She continues by stating that Peirce and mathematicians more generally seek to investigate the “real potential world” of possibilities.  But how can this investigation be undertaken?

            Hull suggests that Peirce may have formulated a mathematical answer to the question of true continua in his 1908 “Neglected Argument for the Reality of God.”  In developing this argument, Peirce integrates his recent insights into the nature of phenomenology and mathematical hypothesis.  The suggestion that the neglected argument is mathematical has, to this point, been almost uniformly dismissed.  In this respect, Hull assumes the position of contrarian, arguing that the proof for the existence-reality of God is a mathematical proof, and, more particularly, a proof for the existence-reality of true continua.  She indicates that this argument can be best understood via Peirce’s work on existential graphs, which hit its stride around 1902.

            According to Hull, and Peirce himself, inquiry—mathematical and otherwise—begins with experience.  Peirce restates this at multiple points, most notably in 1903 by reformulating Aristotle’s comment that “nothing emerges in meaningful conception that first does not emerge in perceptual judgment.”[28]  Carl Hausman echoes this point, writing that according to Peirce “all the sciences of discovery, among which he includes mathematics, rest on observation (1.239-40).”[29]  This form of observation is described as phenomenology later in 1903 when Peirce suggests that “we have to open our eyes and look well at the phenomenon and say what characteristics that are never wanting in it.”  The aim of phenomenology is to muse on an array of phenomena and to witness the appearance of structures common to these phenomena.  Phenomenology is an active seeing or

study which, supported by direct observations of phanerons and generalizing its observations, signalizes a very broad classes of phanerons; describes the features of each; shows that although they are inextricably mixed together that no one can be isolated, yet it is manifest that their characteristics are quite disparate; (and) then proves beyond question that a certain very short list comprises all of these broadest categories of phanerons that are…[30]

 

A few points stand to be drawn out of this passage.  First, phenomenology is a form of direct observation of the phanerons.  Here one must be careful not to suggest that direct observation can seize the Phaneron in its totality, as it stands as the collective whole that could be present in the mind.[31]  Second, this direct observation is at once a “generalization” that “signalizes” classes within the phenomenological field.  It is through this observation-generalization that one arrives at the classes or categories of Peirce’s system.

This relatively early insight into the phenomenological character of the categories appears again in Peirce’s development of mathematical diagrams that simulate the phenomenological method.  These diagrams provide a formal and indirect schematic of the classification of phaneron.  In the last of his Cambridge lectures, Peirce suggests that such graphs can indirectly stand for the true continua of quality that is experienced in any phenomenological musing.[32]  Here we can return to, and complete, Peirce's comment, “Mathematical truth is derived from observation of creation of our own visual imagination…which we may set down on paper in form of diagrams.”

While inquiry begins in musing about the multitude of actual and possible phenomena, it takes flight in the creative play of the imagination that infers certain resemblances beneath the field of phenomenological possibility.  Peirce begins to hit upon this peculiar type in inference in 1893, commenting that “as experience clusters certain ideas into sets, so does the mind too, by its occult nature, cluster certain ideas into sets.”[33]  While this “occult” operation of the mind may not be deductive, Peirce comments that this process of seeing resemblances ought to be considered as a form of logical inference and rests at the heart of diagrammatic experimentation.  The musement of the imagination, in light of phenomena, gives rise to a form of instinctive-intuitive reasoning that exposes the possible relations between the individuals of the multitude.  More simply, yet more impressively, this occult power has the intuitive ability to move from a given set of phenomena to a more holistic conception, a conception of which the given collection is but a part.  Peirce repeatedly employs diagrams (of broken stars, of incomplete circles, of fragmented boxes) to illustrate our power to imagine the possibility of a whole from a fragment.  Mathematicians make the intuitive leap from several elements to the set of which the elements are members, or from concrete examples to general rules.  Mathematical intuition allows mathematicians to infer the possible reality of more complex sets from the existence of less complex collections.

            This non-deductive, intuitive form of reasoning could also be described as a type of Peircean abduction.  More strongly put, we identify mathematical intuition with the abductive creation of new hypotheses.  Without delving into the details of abduction, it is sufficient to note that this inference takes the form of a hypothesis in reference to the possible relations that emerge in the multitude of experience.[34]  Peirce envisions abduction as the way in which these hypotheses are formed and repeatedly underscores the fact that only this novel guesswork can supply novel insight into the possibility of the continuous field of phenomena.  Abductive insight recommends a course of future action and encourages an investigator to move forward on an experimentation of a given diagram.  It is our suspicion that this “moving forward” on a diagram, this “moving forward” in abductively prompted action, captures the essence of true continua.  This is not to say that it captures the totality of true continua but rather that intuitive hypotheses take account of true continuity in its principle of futurity.  Mathematical intuition “moves forward,” inferring the possibility of continuous set expansion into a presumably infinite future.  The character of futurity and the principle of expansion-growth are constitutive of true continua and diagrammatic thought.[35]

Diagrammatic signs, for one, are triads or icons which exhibit “a similarity or analogy to the subject of discourse.”  This similarity or analogy is crucial since the diagram may be finite while its object is infinite.  When the diagram's object is infinite, it cannot be apprehended in its totality except by observing an analogous, finite structure upon which our intuition acts.  Moreover, this resemblance or similarity is in terms of rational relations between the object's parts.[36]  Thus, as Peirce states, “The pure Diagram is designed to represent and to render intelligible, the Form of Relation merely.”[37]

A diagram is a construction that is defined by one or more abstract precepts, and it is on this basis that Peirce equates a drawn geometrical diagram with “an array of algebraical symbols.”  If the diagram is literally drawn or written, “One contemplates the Diagram, and one at once prescinds from the accidental characters that have no significance.”  Once laid out, the construction offers a mathematician the opportunity to see relations “between the parts…or array that are not explicitly expressed by the abstract precept(s).”[38]  For example, the diagram of the positive integers written one after another follows the abstract precept of “successor,” and from that precept one can “skip” every successor and obtain the formal concept of the even numbers.  These new relations are always obtained by rule-bound transformation, which is what Stjernfelt calls the “defining feature of the diagram...[and which make] it the base of [thought-experiment], ranging from routine everyday what-if to scientific invention.”[39]

Even finite objects like triangles must have some generality, for they must apply not only to the concrete diagram that has been written on a blackboard, but must apply to all triangles generally.  The diagram is a schema that is applicable to the infinite collection of empirical triangles.  It is in this form of generalization that reason disclosed the real possibility of true continuity.  True continuity, according to Peirce is “an indispensible element of reality, and that continuity is simply what generality becomes in the logic of relatives, and thus, like generality, and more than generality, is an affair of thought.”[40]  Henry Wang underscores Peirce’s statement that “generality, therefore, is nothing but a rudimentary form of true continuity. Continuity is nothing but perfect generality of the law of relationship.”[41]

The intuitive reasoning that witnesses this generality, this “rudimentary form of true continuity,” is not discrete in the sense of advancing from the abstract precepts that define its logical structure.  Rather, the mathematician goes beyond the given precepts in developing hypotheses that expose the possible relations contained in the mathematical diagram.  The mathematician does not prove this generality, but rather experiences it in intuition.  Hull and others suggest that these hypotheses ought to be considered “new intuitive idealizations,” as products of the creative imagination.  It is in this sense that Beverly Kent suggests, “Peirce attributed creative thinking to the mental manipulations of diagrams.”[42]  It is worth noting that these mental manipulation have a meaning that goes beyond the revision of particular sign configurations on the basis of enumerated rules.  These manipulations are not the products of some brain in a vat, but are embodied and sensuous manipulations.  Diagrammatic manipulations accompany an on-going experience, require an anticipatory involvement, and demand a particular bodily attentiveness.

 

Conclusion

In his search for the foundation of mathematics of true continua in the early 1900s, Peirce’s attention comes to rest on a type of intuition, exemplified in diagrammatic thought that furnishes the experience of mathematical generality and the futurity of continua.  The truth of generality (or thirdness) is not proved, but immediately felt.[43]  By dividing mathematical reasoning into three distinct forms of hypotheses, Peirce avoids the earlier frustrations that he faced in seeking a mathematical conception of true continua.  He realizes that “discrete” mathematics can account for only the first two forms of mathematical reasoning concerning finite and infinite collections.  The third form, concerning true continua, on the other hand, must be produced in the processes of topical geometry.  We have argued that Peirce understands topical geometry in terms of the reciprocity between phenomenology and mathematics.  He underscores the way in which geometrical-diagrammatic hypotheses depend on a type of intuition or insight that allows a mathematician to intimate the principles of true continuity. 

           

           


 

[1]              Potter, Vincent and Paul Shields.  “Peirce’s Definitions of Continuity.” Transactions of the Charles S. Peirce Society.  13,1. 1977.  p. 30.

[2]              Myrvold, Wayne C.  “Peirce on Cantor's Paradox and the Continuum.”  Transactions of the Charles S. Peirce Society.  16, 3.  1995.  pp. 508-541.

[3]              Ketner, Kenneth and Hilary Putnam.  “Introduction.”  In Reasoning and the Logic of Things.  Cambridge, Mass.: Harvard University Press, 1992.

[4]              Herron, Timothy.  “C. S. Peirce's Theories of Infinitesimals.”  Transactions of the Charles S. Peirce Society.  38,3.  1997.  pp. 590-645.

[5]              Hudry, Jean-Louis.  “Peirce's Potential Continuity and Pure Geometry.”  Transactions of the Charles S. Peirce Society.  40,2.  2004.  pp. 229-243.

[6]              Parker, Kelly.  The Continuity of Peirce's Thought.  Nashville, Tenn.: Vanderbilt University Press, 1998.  p. 87.

[7]              Hudry, Jean Louis.  “Peirce’s Potential Continuity and Pure Geometry.” Transactions of the Charles S. Peirce Society.  40,2.  2004. p. 236.

[8]              CP 6.123.

[9]              Ibid.

[10]             Cantor insists that continuity can be best explained through a one-to-one correspondence with the real numbers.  Cantor also characterizes continuity through his concept of “perfectness” and “concatenation,” described by Peirce in the Century Dictionary in 1889.  CP 6.164. 

[11]             CP 1.618 (1898).

[12]             Peirce, Charles S.  Reasoning and the Logic of Things.  Kenneth Ketner, ed.  Cambridge, Mass.: Harvard University Press, 1992.  pp. 159-160 (1898).

[13]             Ibid., p. 248.

[14]             Reasoning and the Logic of Things, pp. 159-60.

[15]             This point is made explicit by Carl Hausman in Hausman, Carl. “Infinitesimals as the Origin of Evolution” in Transactions of the Charles S. Peirce Society.  34.3, 1998.  p. 635.

[16]             Peirce, Charles S.  New Elements of Mathematics.  Carolyn Eisele, ed.  The Hague: Mouton, 1975.  Vol 4, p. 343 (1898).

[17]             CP 4.219 (1897).

[18]             CP 6.188 (1898).

[19]             CP 2.77 (1902).

[20]             CP 1.283 (1902).

[21]             CP 4.307 (1902).

[22]             This example is similar to Peirce’s courier example in the third chapter of the Minute Logic, CP 4.251 (1902).

[23]             Kent, Beverly.  Charles S. Peirce: Logic and the Classification of Science.  Montreal: McGill University Press.  1987.  

[24]             Hausman, Carl.  Charles S. Peirce’s Evolutionary Philosophy.  Cambridge: Cambrdige University Press.  1993.  p. 217.

[25]             CP 5.40 (1903).

[26]             CP 2.197 (1902).

[27]             Ibid. p. 497-498

[28]             Peirce translates the  Latinized form of Aristotle’s comment: Nihil est in intellectu quin prius fuerit in sensu. The Essential Peirce: Selected Philosophical Writings, vol. 2.  ed. N. Houser and C. Kloesel.  Bloomington: Indiana University Press, 1992. p. 227

[29]             Hausman, Carl.  Charles S. Peirce’s Evolutionary Philosophy.  Cambridge: Cambrdige University Press.  1993.  P. 115.

[30]             CP 1.286 (1904).

[31]             CP 7.219 (1908).

[32]             CP 6.204 (1898).

[33]             CP 7.392 (1893)

[34]             Peirce supplies a particularly good example of abduction: given that there is a bag of white beans beside me, and given that I have some white beans in my hand, abduction supplies the hypothesis that the beans in my hand came from the bag.

[35]             For an excellent and detailed exposition of the semiotics of diagrams, see Stiernfelt, Fredrick.  Diagrams as Centerpiece of a Peircean Epistemology.  Transactions of the Charles S. Peirce Society.  36,3. 2000.  pp. 357-384. 

[36]             Stjernfelt, p. 365.

[37]             NEM, p. 59.

[38]             CP 2.216 (1901-02).

[39]             Stjernfelt, p. 370.

[40]             CP 5.436 (1905).

[41]             Wang, Henry.  “Rethinking the Validity and Significance of Final Causation.”   Transactions of the Charles S. Peirce Society.  41,3.  2005.  p. 616.

[42]             Beverly Kent, “The Interconnectedness of Peirce’s Diagrammatic Thought,” in Studies in the Logic of Charles Sanders Peirce, Indiana University Press: Bloomington, 1997, pp. 445-459.  P. 445.

[43]         A decade later, in his Das Kontinuum,  mathematician Hermann Weyl (1885-1955) abandons the atomistic notions of continua and arrives at a similar intuitive position.  Similarly, he suggests that insight rather than proof can secure the foundations of mathematics and, more specifically, a mathematics of continuity. He writes:  “In the Preface to Dedekind (1888) we read that ‘In science, whatever is provable must not be believed without proof.” This remark is certainly characteristic of the way most mathematicians think. Nevertheless, it is a preposterous principle. As if such an indirect concatenation of grounds, call it a proof though we may, can awaken any belief apart from assuring ourselves through immediate insight that each individual step is correct. In all cases, this process of confirmation—and not the proof—remains the ultimate source from which knowledge derives its authority; it is the experience of truth.”    In The Continuum: A Critical Examination of the Foundation of Analysis, trans. S. Pollard and T. Bole. Kirksville, Mo.: Thomas Jefferson University Press. (English translation of Das Kontinuum, Leipzig: Veit, 1918.) 1987.  p. 119.