Continua and clustering processes

1 Introductory considerations

In the last part of his life, Peirce developed his topological analysis of continuity with the purpose of answering some of the questions that emerged in his previous works. In an unpublished work written in 1904, Peirce distinguishes topical geometry (topics) from metrics and graphics (projective geometry). He writes, “In this field of thought we still suppose objects to move about in Space. But we suppose that, at will, any of these objects can be made to expand, to contract, to bend, to twist, and in sort to move free from any law, excepting only that it is nowhere to be broken or welded […]” (Peirce 2010b: 122). And in the Century dictionary, speaking of “Topics” Peirce writes that it is “[t]he most general, fundamental, and naturally elementary branch of geometry, which neither considers lengths, areas, or volumes in their character of being measurable, nor distinguishes straight from curved or crooked lines, nor place from curved or bent surfaces, but studies only the manner in which the parts of places are continuously connected” (Century dictionary vol. IV: 1360).

In the second volume of New elements of mathematics, he similarly defines topology as “the study of the manner in which places are intrinsically connected irrespective of their optical and metrical relations” (Peirce 1976: 165). These definitions show that Peirce’s idea of topology is strictly related to the connection models of the parts of continua and to the conditions of belonging to a continuum. In Peirce’s classification of continuity, Havenel says that in the last part of Peirce’s thought, topology is particularly linked to the analysis of continuity, because it is “the only abstract geometry which purely deals with property of continuity and discontinuity” (Havenel 2008: 120). According to Havenel, Peirce uses topology in order to study two different aspects of continuity. The first, which Havenel calls external continuity, “deals with properties shared by objects belonging to the same class.” The second, which Havenel calls internal continuity, “deals with the mode of immediate connection of the parts of a continuum” (Havenel 2008: 120).

If not connected to the basic questions topology tries to answer, topological analysis may appear as the search for a hidden structure that keeps reality together; something to which reality must be traced back to be understood. This hidden structure is the continuum conceived as a pre-condition that justifies the unity of reality. On the contrary, if we reconnect Peirce’s idea of continuum to the questions it tries to answer, this continuum appears no longer as the condition of possibility for the relations that constitute the unity of reality. On the contrary, it appears as a form whose formation itself is conditioned by relations.

2 The change in Peirce’s idea of form in 1865

The concept of form has an important role in the road toward Peirce’s analysis of continuity. Studying Kant’s thought, Peirce develops a new idea of form, which is clearly expressed in his 1865 Harvard Lecture on Kant. In many works of the same year, Peirce apparently uses the notion of form in a classical sense, as opposed to matter. For example in Unpsychological view of logic he writes:

Every phenomenon is in the first place an image; so that it may be considered to be or to contain a representation. In the second place, the phenomenon may be objectified, or looked upon as a reality; in this way it is said to be or (more usually) to contain matter. For matter is that by virtue of which anything is. In the third place, the differences of its parts and its qualities may be considered, and in this point of view, it is said to be or (more usually) to contain form. For form is that by virtue of which anything is such as it is. Corresponding to the matter of phenomena we have the supposition of external realities or things; and corresponding to the form of phenomena we have qualities. (W1: 307)

Defining matter as “that by virtue of which anything is,” and form as “that by virtue of which anything is such as it is,” Peirce recalls the classical definition of matter and form, considered as real determinants of things, as causes of what a thing is and of how a thing is. However, this classical definition of form is here already modified, because Peirce says that the matter of phenomena corresponds to the supposition of external realities, whereas the form of phenomena corresponds to the process of quality attribution in which phenomena are distinguished and connected. This means that Peirce doesn’t consider matter and form as having to do with the noumenon, but that he rather considers the phenomenal aspect of matter and form.

In Peirce’s works of these years, this modification of the idea of form, which has to do with a consideration of its phenomenal aspects, is deeply related to the definition of symbol. Both in Unpsychological view of logic and in the 1865 Harvard Lectures, Peirce defines symbolization as one of the three possible kinds of representation. Peirce distinguishes between Symbols, Marks, and Analogues, using the structures of denotation and connotation. Mark is defined as a representation that denotes without connoting (this is, for example, the case of proper names). Analogue is defined as a representation that connotes without denoting (this is, for example, the case of an image that can refer to many similar things). Symbol is a representation that denotes by connoting (see: W1 308). In the tenth Harvard lecture, Peirce substitutes the word Mark with Sign and the word Analogue with Copy, but defines Symbol in the identical way: “Every symbol denotes by connoting. A representation which denotes without connoting is a mere sign. If it connotes without thereby denoting, it is a mere copy” (W1: 272).

The main argument of the second Harvard lecture is the logic of science. In this lecture Peirce dwells on symbolic representation because he states that logic deals exclusively with this sort of representation. By defining symbolic representation as what denotes by connoting, Peirce brings into play the concepts of matter and form classically defined, respectively, as what by virtue of which a thing is and what by virtue of which a thing is such as it is. In the 1865 eleventh Harvard lecture, Peirce defines connotation as the relation between a symbol and the form of the object, clarifying that only thanks to this reference to the form (connotation) can a symbol be applicable to a thing (W1: 287).

Peirce writes:

Thus, no matter how general a symbol may be, it must have some connotation limiting its denotation; it must refer to some determinate form; but it must connote reality in order to denote at all; but all that has any determinate form has reality and thus reality is a part of the connotation which does not limit the extension of the symbol. (W1: 287)

The definition of symbol in the works of 1865 involves this strong relation between connotation and denotation. For this reason, as claimed by Peirce in the second Harvard lecture, symbol implies, at the same time, a symbolization of three objects. These three objects are: the possible thing, the possible form, and the possible symbol (see W1: 183). Peirce links these three symbolizations with the three inferences described at the beginning of the second Harvard lecture: Inductive inference is applicable to the symbolization of thing; a-priori inference is applicable to the symbolization of symbol; a-posteriori inference is applicable to the pure form (W1: 184). In the process of the symbolization of form, thus emerges what Peirce calls pure form. Pure form is obtained by an inferential process, which leads from determined to determinant. In Unpsychological view of logic, Peirce argues that pure form is a mental representation in which “the connection of things is eliminated from qualities” (W1: 307). Pure form is something different from form as real determinant of object. Pure form is not what an object is or contains as cause of its how. Pure form is a mental representation, a symbolic representation, that is to say it is a pure fiction obtained by the elimination of the relation to thing (i.e. by that process that some years later Peirce will call prescision). Therefore, in the works of 1865, whereas the relation with matter is considered as a “legitimate hypothesis,” the relation with pure form is considered as fictional. Peirce writes:

Corresponding to the matter of phenomena we have the supposition of external realities or things; and corresponding to the form of phenomena we have qualities. Of these, representation is not altogether hypothetical since we have at least something precisely similar in consciousness. Things are legitimate hypotheses […]. Qualities are fictions; for though it is true that roses are red, yet redness is nothing, but a fiction framed for the purposes of philosophizing […]. (W1: 307)

Some years later, in A new list of categories, Peirce argues that this fictionality of form does not mean that the pure form is arbitrary (W1: 52). This fictionality of form eradicates the idea that the relation to a quality is an immediate impression. The relation to a quality becomes thus mediated by an a-posteriori inference, which produces what Peirce calls fictions.

The fact that pure form represents the connotative part of a symbol means that pure form has an iconic nature (see W1: 467–68). It is important, however, to notice that this iconicity of pure form is strictly connected with its fictionality. For this reason, in the works of the second half of the 1860s, Peirce floats between an idea of icon founded on resemblance (for example when he defines the purely connotative representations as analogous or copies) and a notion of icon whose relation with the object is defined as accordance or correspondence. This second idea of icon clearly appears in Peirce’s definition of validity of representations in the Harvard lectures of 1865. Here he defines the concept of validity by comparing the relation between an object and its color with the relation between a banknote and its value.

The term “objective validity” requires some explanation. Validity is legal tender. Greenbacks are not true cash but they will buy proportionally to cash because they are valid. In the same way, color is not true of objects because it is an affection of the mind and cannot be in matter, but its modifications are true because they correspond to modifications of things. That therefore is valid whose modifications are true (W1: 248).

Despite its fictionality, form is thus valid because it lets us catch true distinctions, true connections, and true modifications of something. All these elements prepare the modification of the idea of icon that, as argued by Nöth in Three paradigms of iconicity, will characterize Peirce’s works of the 1890s.

Nöth (2015:16) writes:

From the 1890s on, Peirce defines the icon as a sign representing its objects by qualities of its own and not by qualities that it has in relation to something else. Similarity is now only a secondary criterion. The icon is defined as having “no dynamical connection with the object it represents”, and Peirce adds, “it simply happens that its qualities resemble those of the object, and excite analogous sensations in the mind for which it is a likeness” (CP 1.566, 1893). Although there is also a dyadic relation between the sign and its object in the definition of the icon, this relation is not “dynamical”, and its “relative character” (CP 1.566, 1893) is weak. More precisely the icon is a sign by “virtue of its internal nature” (CP 8.335, 1904) or because “a quality that has qua thing renders it fit” to function as a sign (EP 2. 273, 1903).

Underlining this fictionality of form, Peirce definitively displaces the concept of form from its classical formulation. Form is thus no longer considered as the cause of how a thing is, i.e. its real determinant, but becomes a logical determinant, i.e. the result of an inference. In the second 1865 Harvard lecture, Peirce shows how this deep change in the idea of form happens in Kant’s thought, in the passage from his early works on the first Critique.

Accordingly, Kant in his first essay on this subject, which was published 12 years before the Critique, uses matter and form for the effect of material and formal causes. That which the world is (abstracted from how it is) is its elementary parts. How the world is (abstracted from its existence) is the coordination of those parts or their relation to each other as parts, potential or actual. (W1: 250). But, in the Critique of pure reason Kant proposes a radically new idea of form and matter, because they are no longer conceived as causes or real determinants of what and how something is, but as logical determinants of what and how something is.

Peirce writes:

But in the Critique he again modified the meaning of the terms. Instead of making the parts and coordination merely subjective he regards them as belonging to the immediate object of perception antecedent to thought, which he calls the appearance. This approximates to the original meaning again. For as originally matter and form were the real determinants or causes of the that and the how; so with Kant they become the logical determinants or reasons of the that and the how. The Matter is that in the appearance which corresponds to the impression upon the sense; for without sense no mental representation could exist. The Form is the condition of the possibility of the relations of the elementary parts of the representation. (W1: 250)

In Peirces Deconstruktion der Trazendentalphilosophie, Thomas Hünefeldt shows how in this passage Peirce distinguishes three different conceptions of form and matter. In the first, matter and form are considered, in accordance with their classical definition, as the causes of the that and the how; in the second, formulated by Kant twelve years before the first Critique, matter and form are considered effects respectively of material and formal cause; in the third, formulated by Kant in the first Critique, matter and form are considered neither as cause nor as effects, but rather as reasons and more precisely as reasons of the that and the how (see Hünefeldt 2002: 241) .

Underlining this mutation of the idea of forms in Kant’s thought, Peirce highlights the displacement from form as cause to form as reason. This displacement makes possible a new definition of form, which becomes “the condition of the possibility of the relation of the elementary parts of representation” (W1: 250).

In Kant’s first Critique, according to Peirce, the concept of form is thus no longer conceived as the cause of being, that is to say as what makes the things how they are, blocking them in their own being. Form becomes, on the contrary, a dynamic principle of coordination between the elementary parts of representation. Space and time are in Kant’s first Critique the condition sine qua non of this coordination.

In the eighth 1865 Harvard lecture, Peirce defines Thing, Form and Representation:

I however would limit the term neither to that which is mediate nor to that which is mental, but would use it in its broad, usual and etymological sense for anything which is supposed to stand for another, and which might express that other to a mind which truly could understand it. Thus our whole world – that which we can comprehend – is a world of representations. […] The thing is that for which a representation might stand prescinded from all that would constitute a relation with any representation. The form is the respect in which a representation might stand for a thing. (W1: 257)

Form becomes what interrupts a simple biunique relation between thing and its representation. Representation presupposes a form, presupposes a “respect in which” it stands for a thing. This “respect in which” is what, some years later, Peirce will call ground.

In a manuscript written in 1897 Peirce writes, “The sign stands for something, its object. It stands for the object, not in all respects, but in reference to a sort of idea which I have sometimes called the ground of representamen” (CP 2: 228). Interpreted as reason rather than as cause, form becomes a productive event, which lets things enter in an interpretative process (see Huslwit 2002).

3 The alteration of timeline in the interpretative process

The new conception of form as reason is the starting point of Peirce’s work on categories in the second half of the 1860s. Thanks to this new idea of form, the problem of categories becomes the question of the coordination of the elementary parts of representation, that is to say the question of the passage from the multiplicity of perception to the unity of representation. In Peirce’s work the problem of categories is the problem of the passage from multiplicity to unity.

In his essay of 1867 titled On a new list of categories, the question of form begins to be interwoven with the one of continuity. The passage from multiplicity to unity cannot be reduced, as stated by Peirce, to a mere sum of element.

In this significant essay, categories are interpreted as connective moments of an inferential process that has to be continuous in order to be valid. The work on categories is thus the work on a transformative movement, on a process in which a multiplicity of sensorial impressions is reduced to a general unitary representation. Representation is the constitution of a continuous structure, which has in the form, as we have seen above, its condition of possibility. In these years, the aim of Peirce’s work on categories is not the definition of a static structure of rules, nor the construction of a table of categories which explains how inference works mechanically. Peirce’s question about categories is the question about the dynamic process of interpretation. To pose the question about categories, in this sense, is the attempt to discover the laws that regulate the work of the interpretative process, what gives reality form.

In this interpretative dynamic, continuity constitutes the guarantee of the validity of the inference. If the passage from multiplicity to unity is valid, this passage can be represented on a continuous diagram. That’s why for Peirce it is important to build “a list,” and not only a table of categories. If compared to Kant’s theory, the “newness” of Peirce’s categories is not only in their reduced number. The “newness” of Peirce list is that it is a list, and not only a table, like the table of Kant’s categories. A “list” presupposes not only a number of items, but also a criterion of an order between them. A list must have the possibility of using ordinal numbers to give a precise order to the contained items. In Kant’s table of categories, Kant doesn’t indicate any criterion in order to build a list of categories. Through his work on categories, Peirce tries to describe a process. Because of this, he needs a list, that is to say an ordered succession of moments.

Peirce individuates this criterion using the method of prescision. Prescision establishes an order of categories, which makes them not only rules of reasoning but also connective moments of a continuous and ordered process that conducts from substance to being.

In On a new list, Peirce works on categories as connective moments of the process that conducts from multiplicity to unity. But, in the attempt to explicate these connective relations, Peirce uses a twin-track approach, opening two different perspectives. The first perspective is opened by the use of the method of prescision. By prescision Peirce is able to establish an order of the categories, thus making them not only an instrument of a taxonomic classification of perceptions but also connective moments of the continuous and directional process that leads from being to substance (see W2: 50–51). As stated by Peirce, although categories can be analyzed prescinding from their condition of possibility, it is not possible to analyze them prescinding from the condition that makes them thinkable. This mean that, “reference to a ground cannot be prescinded from being, but being can be prescinded from it” (W2: 53); “Reference to a correlate cannot be prescinded from reference to a ground; but reference to a ground may be prescinded from reference to a correlate” (W2: 53); and that “Reference to an interpretant cannot be prescinded from reference to a correlate; but the latter can be prescinded from the former” (W2: 54). As Peirce observes, by prescision it is possible, at the same time, to distinguish, to relate, and to order the categories, because the non-reciprocity of prescision indicates the direction of the process, in which the previous grade is what makes it possible to think the next one.

By prescision, Peirce thus obtains an order between categories, which could be put on a timeline, describing a process which follows a series of steps, a process which follows a continuous and unidirectional schema, thus making possible the “quantitative” transformation from multiplicity to unity, that is to say from substance to being. In Peirce’s account of purposefulness, Gava (2014) argues that Peirce’s method of precision by which he obtains the categories as starting from experience is very similar to the method applied by Kant when he works on the apriority of time or on the unity of apperception. In this sense, as stated by Gava, Kant’s transcendentalism can be considered as a procedure that starts from experience rather than from true and unquestionable premises. For these reasons, Gava asserts that prescision is a transcendental method (2014: 151).

Prescision thus starts from the experience as such, in order to abstract from it the elements that are necessary to the structuring of experience itself. This means that from the perspective of prescision, the passage from substance to being is a process that has already taken place, a process that is already ordered and blocked. Accordingly, the result obtained by the method of prescision is a “schema” of the process. This schema represents the rules, the structure of the passage from substance to being. But, in the same essay and at the same time, as if in two lines of counterpoint, Peirce shows the same process from another perspective. In this second perspective what is at stake is not the conditions that make thinkable the phases of the process, but rather the conditions that make the process possible. As stated by Peirce, the reference to the ground has as its condition of possibility the reference to the correlate, and the reference to the correlate has as its condition of possibility the reference to the intepretant. In this second perspective, representation, i.e. the reference to the interpretant, becomes the condition of possibility both for the reference to the correlate and for the reference to the ground. In this second perspective, the passage from substance to being has already begun, the interpretative process is in act, all is still working. In this “being in act,” the terms of the process begin to refuse a linear and unidirectional temporal structure, because the third moment, the one of representation, seems to precede the first one, in an interpretative process that has already begun. The time of the passage from substance to being seems to twist on itself in a spiral form, in which the third moment seems to come before the first.

In the essay of 1867, Peirce succeeds in keeping together these two perspectives of the passage from substance to being. The perspective obtained by prescision explains that, although quality has as its condition of possibility both relation and representation, once the quality is obtained, I can conceive it prescinding from the condition necessary for its constitution, prescinding from its condition of possibility. This means that if on the one hand it is true that “red” can be thought without the reference to the categories of relation and representation, on the other hand is also true that without the reference to the correlate and to the interpretant, “red” cannot mean anything, that is to say that the passage from the substance to being starts from an interpretant. The definition of a color needs the possibility of connecting the sensation of a color with other sensations of other colors in order to construct a classification, in order to build a science of colors.

These two perspectives thus provide a double view on the process. In the first one, offered by the method of prescision, we have a schema of the process, which shows its rules, whereas in the second one, we have a moving image, which shows the active process in which the passage from substance to being has already begun. In this second perspective the reference to the interpretant is the starting point of the process and the condition of possibility of is functioning. In A new list, Peirce uses a suggestive example in order to explain the necessity of representation, i.e. the necessity of the interpretant as the condition of the possibility of reference to the correlate. This example is correlation of the small letters p and b (W2: 53). To put p and b in relation, I can, for example say that p is an inverted b. But to conceive this relation, I have to represent to myself a rotation on its own axis of p so that it can become a b. This animated representation is the interpretant, which creates an action making the two letters instants of the same rotatory movement, thereby establishing a continuity that puts them in relation. This constitutes the form of the interpretative process.

In A new list, the interpretant becomes the starting point of the interpretation process which allows the passage from substance to being, operating something like a rotation of time in which two apparently different things enter into a relation. Peirce shows that this movement is not a linear, but rather a spiral movement, which never comes back to its starting point. In this spiral the third moment becomes the condition of the possibility of the first one and, for this reason, it activates a process in which the categories become moments of a continuous and unitary movement. This movement is not continuous because it is unitary, but, on the contrary, is unitary because it is continuous. This movement, which starts from an interpretant, is not kept together by the linear and unidirectional succession of time (chronological time).

In A new list, therefore, the image of a linear and unidirectional continuum begins to appear as insufficient to support Peirce’s thought, because the connections that are established by the movement from substance to being, are not representable on a timeline.

This alteration of the timeline is form, exactly as Peirce defines it in the Harvard lectures on Kant, as a “condition of possibility of the relation of the elementary parts of representation.” Form is this modeling dynamic which, starting from interpretation, allows the process of the passage from substance to being to begin.

4 The dynamic continuum of interpretation

As mentioned in the previous section, what characterizes Peirce’s complex work on continuum is the necessity, which appears in A new list, of an analysis of the interpretative process, which is able to go beyond its schematization obtained by the method of prescision. The main point to take from this discourse is that if the interpretative movement is seen while it is happening, as something already begun and never finished, then continuity can no longer be considered as a condition which guarantees the validity of the inferential process. Rather, it has to be considered as what the inferential process itself produces while it is active, in movement, alive.

The two paths on which Peirce’s analysis of temporality moves are: that of an interpretative process in action; and that of the emersion of a non-linear continuity, which cannot be represented by the traditional idea of a linear and unidirectional time. These two paths induce Peirce to introduce, between the 1880s and 1890s, the infinitesimal in his work on continuum.

When in the 1892 essay on The law of mind Peirce explains why it is necessary to introduce the concept of the infinitesimal in the study of continuity, he distinguishes two types of collections: finite collections and infinite collections (see W8: 139 and following). Peirce argues that these two collections are different, because the “syllogism of the transposed quantity” is applicable to the finite collections, but is not applicable to the infinite one.

One of the examples Peirce gives to explain the syllogism of transposed quantity is:

If an insurance company pays to its insured on an average more than they have ever paid it, including interest, it must lose money. But every modern actuary would see a fallacy in that, since the business is continually on the increase. But should war, or other cataclysm, cause the class of insured to be a finite one, the conclusion would turn out painfully correct, after all. (W8: 139–140)

The syllogism of transposed quantity is an inference that extends some properties of a region of a collection to the whole collection. This inference is not applicable to infinite collections, because they, as in the above example, are still growing. If, for example, a race is already finished and I know in how much time a certain amount of space has been covered, then I can calculate the average speed. But, if it were still happening, the average speed would be necessarily partial and not indicative. More indicative would be the instantaneous speed, but I can calculate it only if instant becomes interval or “moment,” as Peirce says (W8: 139). The passage from the instant to the moment takes in itself the idea of a relation that constitutes a continuity. The passage from the instant to the moment has as its consequence the idea that continuity is linked to an expansion state, to the being alive of the collection, to the fact that it is a form that is taking shape.

In The law of mind, Peirce describes infinite collections as changing and “living” collections. These are collections in which the process of formation is always active, already begun and still not finished. Using the concept of the infinitesimal, Peirce is able to describe consciousness as an infinite collection that is not made by isolated points, but rather by passing moments that are limits in a series, in which the present is connected with the past by real infinitesimal steps.

But inside this first continuity that is constituted by passing and interconnected moments, there occurs a second continuity that is constituted by the association of sensation and ideas.

Peirce writes, “Time, as the universal form of change, cannot exist unless there is something to undergo change, and to undergo a change continuous in time, there must be a continuity of changeable qualities” (W8: 146).

This second order of continuity, inside the continuity of the time of consciousness, has to do both with the association of sensation in ideas and with the association of the ideas between themselves. Therefore, what is at stake here is the problem of the constitution of temporal objects. The problem of the constitution of these continuities inside the continuous time of consciousness becomes the main problem in the development of Peirce’s analysis of continuity in the 1890s.

In order to clarify this problem and to show how it acts in Peirce’s thought, I will recall a distinction operated by Husserl, when in the Seefeld manuscripts he works on the constitution of temporal objects in the temporality of consciousness. In these manuscripts Husserl distinguishes temporal continuity from continuity in time (Husserl 1985).

Continuity in time presupposes, according to Husserl, a real identity of a temporal object, as in the case of a sound whose pitch remains the same throughout an interval of time. This sort of continuity is not a problematic continuity, because it has in itself its own justification, which is constituted by a real identity of the temporal object. But, according to Husserl, there are other types of continuity, which he calls temporal continuities. This second type of continuities is not motivated by a real identity, because there isn’t any material aspect that remains unchanged. This continuity is the simple juxtaposition of temporal objects, which are perceived together as a unitary temporal object without a real unity, without a material identity. It is the case of different sounds which compose a melody or which resound in a single chord or in a cluster. In the case of this second type of temporal objects, it is the position, it is the encounter, which generates the sense and determines the continuity, and not the contrary.

In the 1890s Peirce explains the passage from particular feelings to a general idea by the infinitesimal variation of intensity.

In a manuscript written in 1895 titled On the logic of quantity, Peirce writes:

By feeling is here meant that which is immediately present in consciousness. That is to say, it is wholly present in any moment and endures without coming or going. Thus the color of vermillion under a given degree of illumination is a feeling. It has its luminous intensity, its chromatic intensity, and its intensity of specific hue. Red is a more general feeling composed of vermillion and other reds taken together. (Peirce 2010b: 49)

In this sense, as stated by Peirce in The law of mind, feelings are intensive continua, that is to say they are continua obtained by infinitesimal variation of intensity, which give rise to general ideas.

Peirce writes in Logic of quantity, “A finite interval of time generally contains an innumerable series of feelings; and when these become welded together in association, the result is a general idea” (Peirce 2010a: 149).

As argued by Peirce, this general idea is a “living feeling.” This means that this general idea is not a blurred copy of the sensation, but it is rather what lets sensation be alive. In this sense, sensations are activated by an interpretative process, which puts in relation something with something else, establishing a continuity. This intensive continuity could be, recalling Husserl’s distinction, a continuity in time, because the variations of intensity which compose the general idea of something have a real continuity, a material aspect which makes them all variations of the same general idea. Exactly as when we perceive as unitary a sound which remains at the same pitch even though its intensity or its timbre is changing. Therefore, in the case of temporality of consciousness, Peirce resolves the question of continuity by the infinitesimal intervals, and in the case of the associations of sensations he resolves the question of continuity by the infinitesimal variation of intensity.

But there is a more complex and problematic level of continuity that has to be explained: It is the problem of the continuity established by the association of ideas.

In The law of mind Peirce dwells on situations in which an idea can have an influence on another one. In this essay he considers ideas inside the temporality of consciousness, inside a temporal line, so that an idea in the past can have an influence on an idea in the present, which can have an influence on an idea in the future.

But, in the manuscript on Logic of continuity, he looks deeply at the question of the association of ideas. In this manuscript he analyzes not only how an idea can have an influence on another one, but also how ideas are connected with each other in the present moment. In order to define this particular association, Peirce uses, in the manuscript of 1895, the word cluster, clustering. A cluster is an agglomerate in which something is assembled without a justification. The being together of the cluster is an encounter determined by an assumed position, as for example it happens when we hit with an opened hand the keyboard of a piano, producing a sound that is unitary but contains elements which are different and whose relation is not predetermined, but is an accident, a chance.

In the manuscript of 1895, Peirce argues that an association of ideas can be determined by contiguity or resemblance. In both cases, the motivation of the association is what Peirce calls here an “occult power,” which is external in the case of contiguity and internal in the case of resemblance.

Peirce writes, “The clustering of ideas is either due to an outward occult power or to an inward one. That it is due to some occult power is plain from this that the ideas although they are in our minds and thus normally subject to our will, cluster in spite of our will, and that in certain regular ways” (Peirce 2010b: 50).

Peirce specifies that the adjective “occult” has here to be intended in the sense that “nothing about it can be learned by mere observation of phenomena” (Peirce 2010b: 50).Ideas can cluster together because experience keeps them regularly together. It is what Hume calls contiguity.

But, Peirce says, “Is to be observed that the contiguity consists in ideas being brought together in experience, and is not the cause of it. That cause it that occult power acting like our wills, though with far greater might, which lies behind experience, and which the old philosophers called Nature” (Peirce 2010b: 50).

In a similar way, in order to explain the clustering process that is due to an association by resemblance, Peirce says: “As before, it is to be remarked that the resemblance consists in the ideas clustering together […], and is not the cause of the clustering. That cause is an occult power which seems to lie behind the inward world just as Nature lies behind the outward world. It is often called Reason” (Peirce 2010b: 51).

This “occult power” is a modality of the relation between ideas, which is not explicable by what Husserl calls “continuity in time,” because it is an instantaneous relation that produces a “living” and “changing” relation. Peirce writes in The law of mind:

[T]hese general ideas are not mere words, nor do they consist in this, that certain concrete facts will every time happen under certain descriptions of conditions; but they are just as much, or rather far more, living realities than the feeling themselves out of which they are concreted. And to say that mental phenomena are governed by laws does not mean merely that they are describable by a general formula; but there is a living idea, a conscious continuum of feeling, which pervades them, and to which they are docile. (Peirce W1: 154)

This “living reality” mentioned by Peirce is linked to the vitality of the interpretative process, which manifests itself as a process of association of ideas that is impossible to control or to stop. Form, defined since Peirce’s early works as condition of possibility of nexus, becomes, in the 1890s, this occult power, which has the capability of putting things in relation through the form of a cluster. These clustering relations are not produced by a sense but, on the contrary, produce sense.

At this point of Peirce’s work, the problem is that the continuity which is established at the level of the association of ideas does not have the aspect of a general structure but has rather the aspect of an interpreting “living” force. The continuity at the level of the association of ideas thus appears a very different continuity if compared to the infinitesimal continuity of consciousness and to the intensive continuity of feelings. Coming back to Husserl’s distinction, the association of feelings that produces general ideas is a “continuity in time,” but the association between ideas is a “temporal continuity.” This difference is very important, because in the case of the continuity in time, the associative relation is constituted by the structure of continuity, while, in the case of temporal continuity it is the associative relation itself which constitutes the structure of continuity. In this latter case the structure of continuity appears as a living, interpreting, and changing form. Temporal continuity constitutes not-necessary connections, i.e. connections that have not in themselves their own justification. Temporal continuity constitutes a form that can be defined only by its transformations. The reference to the temporality of consciousness is not enough in order to analyze temporal continuities.

It is not enough because the unitary perception of a temporal object cannot be justified from the fact that the parts of a temporal object (for example the sounds of a melody) are caught by the time of consciousness. His research on this particular form of continuity is what brings Peirce, at the beginning of the twentieth century, to develop his topological analysis of continuity, which applies a model of analysis that is more connected with the nature of space than with the nature of time. Topological analysis of continuity considers temporal continuities as forms that are defined by their transformations, thus studying the models of relation and of constitution of continua.

Topological method will deeply change Peirce’s idea of continuum, because it displaces the analysis beyond the merely analytical descriptive level, toward the question of the constitution of continuum. Topological method will realize the movement that, as we have seen above, begins in On a new list of categories, which displaces Peirce’s discourse from the condition of the thinkability of continuum to its very conditions of possibility.

Topological analysis cannot work by dissections, because it works on a moving process, on the living body of an interpretative process. By decomposition and dissection it would not find the relations it looks for, because these relations are activated by the living (dynamic) movement of interpretation. Therefore, topological analysis of continua necessarily has to work by complexification, detecting the relations between a continuum and the continua of a higher grade. Topological analysis will thus open Peirce’s thought toward the idea of an eventual continuity in which form is transformed on the background of more and more general continuities, for which interpretative processes are already in act. This is to say, topology opens Peirce’s thought toward an analysis which does not work on the nexus to catch them in the form of a law, or of a schema, but that studies the nexus which constitute continuity in order to show their infinite potentiality of transformation starting from an untraceable origin.