**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 *s _{n}* =

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 *s _{n}* on (-∞,

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.