Logical Principles of a Topological Explanation

1 Introduction

The aim of this paper is to introduce Peirce not as a semiotician but as a logician whose theories in logic provided new insight into science. Prior to Peirce and his Logic of Relatives, scientific explanations were dealing with substances and attributes, while the new thinking proposed by Peirce is based on relations. ^[1] The kind of thinking and explaining based on relations is extremely important in today’s science and technology (take for instance neural networks). It is widely recognized that Peirce’s philosophy and logic were largely inspired by his early studies of chemistry. ^[2] His understanding of chemical compounds in terms of topology and relations, especially in creating diagrams or schemata of molecular relations, ^[3] and which he later applied to his studies in logic (Ambrosio and Campbell 2017), were crucial elements here. ^[4]

The originality of Peirce’s approach consisted of his observation of molecular relations rather than concrete chemical qualities based on his interest in molecular topology and its spatial representations. Later applied to logic, Peirce’s topological approach led to the creation of his well-known Existential Graphs (EGs). Apart from its historical value, Peirce’s iconic logic is also being studied and reinterpreted in current philosophical research (Shin 2002; Pietarinen 2006; Stjernfelt 2007; Pietarinen and Stjernfelt 2015). The aim of this paper is to reach an understanding of Peirce’s iconic logic in terms of topological explanations so as to introduce Peirce as a precursor and founder of topological thinking in science. In the context of contemporary discussions on topological explanations in the philosophy of science, it is instructive to review Peircean diagrams in light of specific current problems within, for example, the philosophy of life sciences.

Although Charles S. Peirce is known mainly as a semiotician and philosopher of pragmatism, he also made several useful contributions to logic. His work consists mainly of an original elaboration of two kinds of notation of logical relations: the symbolic and the graphical. This paper focuses on the latter, since we recognize a certain continuation of and similarity to the Peircean graphical way of notating logical relations and the current trend in topological explanations. Peirce’s graphical logic is referred to as “iconic logic” in this paper. We use the term “iconic,” borrowing it from the proper terminology of Peirce’s philosophy and semiotics.

According to Peirce, an icon is a type of sign which relates to its object by resemblance. This resemblance can be imagic, with the sign visually resembling its object or it can be relational, with the sign resembling its object by a similarity in internal relations. ^[5] The relational similarity can be represented as a diagram or a schema. Peirce called this kind of iconicity “diagrammatic iconicity.” Thus, Peirce’s iconic logic could also be called “diagrammatic logic.”

Peirce’s diagrammatic logic has been further elaborated or reinterpreted by several philosophers and semioticians (Shin 2002; Pietarinen 2006; Pietarinen and Stjernfelt 2015). The very concept of “diagrammatology” has been analyzed by Stjernfelt (2007).

Peirce’s iconic logic is depicted in the form of EGs. EGs are graphical systems depicting basic logical relations and are tripartite: Alpha, Beta, and Gamma. Simply expressed, because Peirce’s graphical logical system was not completed, we can equate Alpha graphs with propositional logic and Beta Graphs with predicate logic, while Gamma graphs embrace several elements of modal logic. For Alpha graphs, the primitive elements of the graphical logical system are symbols for sentences and cuts, which represent negation. The juxtaposition of graphs is interpreted as conjunction. For example,

Figure 1

Juxtaposition of graphs

If sentence c is “Carl is smiling” and sentence g is “Gustav is angry,” we can interpret the three following graphs in this way: (a) “Carl is smiling and Gustav is angry,” (b) “Carl is not smiling and Gustav is not angry,” (c) “It is not the case that Carl is not smiling and Gustav is not angry.” The last graph also has the interpretation: “Carl is smiling or Gustav is angry,” and thus the Alpha graphical representation of disjunction can be seen.

For Beta Graphs, the primitive elements are symbols for predicates, cuts, and lines of identity. According to Shin:

The Beta system is analogous to a pure symbolic first-order language with an equality symbol, but without a constant symbol. (Shin 2002: 39)

As with Alpha graphs, the cut represents negation, and thus the main difference between Alpha and Beta Graphs is connected to the lines of identity. For the sake of clarity, a brief definition of the line of identity, expressed by Peirce himself, can be proposed:

A heavy line, called a line of identity, shall be a graph asserting the numerical identity of the individuals denoted by its two extremities. (Peirce CP 4.444)

The identity line remains for a relation expressing that the predicates share a common variable. To illustrate, we propose investigating Peirce’s own example (CP 4.442)

Figure 2

Identity line

Beta Graphs in particular are of crucial importance for this paper (see Shin 2002). ^[6] The line of identity with possible multiple branching, which connects more than two predicate symbols, represents an object in an even more complex way. For example:

To interpret Figure 3, we have to posit that G, B, C and M are predicates (G – is a girl, B– is a boy, C– is from the House of Capulet, M – is from the House of Montague) and L is a binary relation (L – is in love with –).

Figure 3

Beta Graphs of the relations in Shakespeare’s Romeo and Juliet

As Shin has already pointed out,

This device, representing sameness iconically, makes a clear distinction between Beta graphs as a graphical system and other predicate languages as symbolic systems. While in a symbolic system tokens of the same type of letter represent the same individual that each token denotes, in Peirce’s Beta system a network of lines represents the same individual denoted by each branch of the network. That is, Peirce’s system graphically represents numerical identity with one connected network. (Shin 2002: 54)

We can conclude our introduction to Peirce’s EGs with a final example, the interpretation of which is left as a challenge to the reader.

Figure 4

Beta Graph of famous physical theories

As in the previous example: A, M, E, and P are predicates (A – is without unique axiomatization, M – is anticipated by Mach, E – is established by Einstein, P– is anticipated by Planck) and C, V, and O are binary relations (C – is compatible with –, V– is a simplified version of –, O– is in opposition to –).

This paper assumes the following scope.

1. We introduce the notion of Peirce’s NonReduction Theorem, explicating the understanding of triadic predicates by means of the Peircean iconic (diagrammatic) logical system.

2. In connection with the previous section, we compare dyadic and triadic conceptions of logical relations in the specific context of biology. The notion of topological explanation is also designated.

3. We expand the notion of NonReduction Theorem to a more general philosophical context, namely to the context of liberal naturalism within the philosophy of science. As an empirical example, we apply the aforementioned theories to several areas of current biological research, focusing on proteomics and the protein folding process.

The overarching aim of this paper is to apply Peircean iconic logic, specifically EGs to contemporary debates on topological explanations in the philosophy of science.

2 The reduction thesis and existential graphs

Although Peirce never explicitly formulated the so-called NonReduction Theorem or Reduction Thesis, many Peircean scholars widely use this term (e.g. Burch 1997; Ketner et al. 2011). Peirce has nevertheless clearly described the idea of (non-)reduction in a number of places in his work. Consider the following:

For were every element of the phaneron, a monad or a dyad, without the relative of teridentity (which is, of course, a triad), it is evident that no triad could ever be built up. Now the relation of every sign to its object and interpretant is plainly a triad. A triad might be built up of pentads or of any higher perissad elements in many ways. But it can be proved – and really with extreme simplicity, though the statement of the general proof is confusing – that no element can have a higher valency than three. (Peirce CP 1.292) ^[7]

The (non-)reduction, or the teridentity of relations, is central to all Peircean thinking. This is probably why Peirce himself never formulated it as a theorem, since it is implicit throughout his entire work. Nevertheless, (non-)reduction is also stated explicitly (see the quotations above) and is exhaustively explicated in the Logic of Relatives in terms of the distinction between genuine and degenerate triads (CP 1.363). We will focus on (non-)reduction as present in the Logic of Relatives, since it is the relation that plays a crucial role in contemporary topological explanations. First, we consider certain influential objections to Peirce’s triadic relation to establish if this theory holds in the face of several “orthodox” logicians’ refutations.

There is an ongoing discussion about the possibility of reducing the triad to dyadic relations. ^[8] Many researchers, in contrast, have upheld the coherence of Peirce’s triadic predicate logic (Burch 1997; Ketner et al. 2011). One main objection against Peirce’s triadic predicate logic consists in the difficulties arising when trying to express it in terms of symbolic logic. Symbolic logic, like any other kind of symbolic notation, natural language for example, is limited to, but equally defined by, its linearity. This means it is limited to a unidimensional and unidirectional script based on relations of simple juxtaposition. Yet the essence of Peirce’s teridentic relation resides in its ontological trifold status beyond juxtaposition (coexistence).

The essence of the teridentic relation lies in its ontological status, which is not expressible linearly. The limits of juxtaposition are described in Brunning (1997). This is probably one of the reasons that led Peirce to elaborate his EGs, a non-linear formalization of logical relations based on iconic (diagrammatic) representations. We focus on teridentity as an identity line in Beta Graphs that connects the ends of a triad.

The aforementioned limitation of symbolic logic for expressing teridentic relations does not, however, presume that these relations are inexpressible by means of a linear script. Peirce himself proposed a purely symbolic notation to demonstrate the teridentity relation and expressed his (non-)reduction thesis by means of algebra (Peirce 1897), with this idea being further discussed by Legg (2012).

Let us examine Beta Graphs more closely. Contemporary interpretation of Beta Graphs embraces two primary directions:

1. that of Zeman (and others), who translate Beta Graphs into symbolic logic;

2. that of Shin, who highlights the role of Beta Graphs because iconicity facilitates direct interpretation of the logical relations.

Beta Graphs are more than a device to facilitate the interpretation of logical relations; they embrace additional information and are not mere illustrations of algebra or symbolic logic. Although it is always possible to translate one system into another, this does establish their relevance. The crux is the extra value Beta Graphs contribute. We identify two main reasons that uphold the creation and elaboration of EGs.

EGs reveal something that is not immediately or completely apparent, namely that formal logic does not have to join a specific symbolicity (arbitrariness) or linearity. In other words, symbolic logic as we know it is not the only possible way to formalize logical relations. This might seem axiomatic, yet Peirce’s attempt to construct a non-symbolic logical notation based on iconic representations is, we assume, an important milestone in the history of diagrammatic logic. Other attempts to propose an elaborated iconic logical notation have appeared post Peirce and take the form of a reaction to, or a continuation of, Peirce’s logical system (Shin 2002; Pietarinen 2008).

The second reason is that Beta Graphs, in contrast to symbolic logic notation, work perfectly as schemata to demonstrate and fortify the NonReduction Theorem. The impossibility of reducing triadic relations to dyadic or monadic ones cannot be wholly demonstrated linearly. Unfortunately, Peirce’s work on iconic logic was never completed, and we can only speculate about several possible continuations of work on EGs (Shin 2002).

The so-called NonReduction Theorem is often associated with the Relational Completeness Theorem. Indeed, many authors associate the two theorems, which is explained by the two theorems’ interconnection, yet putative interdependence. Ketner formulates both theorems as follows:

(I) Triadic relational types cannot be constructed using bonding and a resource base consisting only of monadic and/or dyadic relational types (NRT); and

(II) Relational types of any valency can be constructed using bonding and a resource base consisting of monadic, dyadic and triadic relational types (RCT). (Ketner et al. 2011: 9)

Burch, in contrast, subsumes both theorems in one thesis, the so-called Reduction Thesis, in which NRT and RCT are present in the form of positive and negative components.

Peirce’s Reduction Thesis is a doctrine that has both a negative and a positive component. The negative component of the Thesis says, first, that relations of adicity 2 may not in general be constructed from relations exclusively of adicity 1; and, second, that relations of adicity 3 or greater may not in general be constructed from relations exclusively of adicities 1 and/or 2. The positive component of the Thesis says that all relations, regardless of the domain—of arbitrary (non-negative integer) adicities—may be constructed from relations exclusively of adicities 1, 2, and 3. (Burch 1992: 670)

The RCT concept suffuses all Peirce’s thinking. Apart from its connection to the NonReduction Theorem, relationality in general constitutes the core of all of Peirce’s philosophy. The semiotic theory of continuum (synechism) is a concept encompassing both Peircean semiotics and mathematics and is closely related to the topological understanding of geometric relations.

Although the term semiotic continuum as presented by Peirce is not synonymous with the mathematical notion of continuum, it is clear that Peirce as a semiotician was influenced by his keen interest in mathematics. Even outlines of modern mathematical concepts can be found in Peirce (e.g. Hudry 2004). In semiotics, synechism is a whole of interrelated signs composed of representamen, object, and interpretant. Interpretants guarantee a potential relation of one sign with others, in this way creating infinite semiosis and the continuum of signs. The semiotic definition of synechism reflects the RCT in that the infinity of the whole semiotic continuum is always generated by triadic signs.

The theory of synechism has been re-elaborated by modern semiotics, for example by Eco (2007) who uses the concept of encyclopedia instead of synechism , which can be graphically represented by a labyrinth. A semiotic labyrinth is simply a web of interrelated signs.

We previously assumed that Peircean (non-)reduction thinking might be one reason for Peirce’s elaboration of the iconic (diagrammatic) logic (EGs). The very notion of (non-)reduction somehow instigates a problematic linear representation: linear script is by its very nature reduced to one-dimensional space. Here we should specify the diagrammatic character of EGs.

Peirce’s celebrated classification of signs distinguishes between three types of signs: indexes, symbols, and icons. An icon is a sign that refers to something else based on its resemblance. Icons are further classified into images, metaphors, and diagrams. Imagic iconicity is based on mere likeness; metaphoric iconicity is also based on likeness, but it has to be conventionalized by means of usage. Finally, diagrammatic iconicity is based on resemblance in the structure of internal relations. The crucial difference between diagrams and the other two types of icons is that diagrams do not resemble their objects; it is only in respect to the relations of their parts that their likeness commingles. In other words, we conceive diagrammaticity as a topological type of iconicity.

In general, the use of diagrams instead of linear representations was typical of Peirce, whose work was oriented on seeking diagrammatic resemblances. We have already made reference to Peirce’s inspiration derived from chemistry for his work on the Logic of Relatives. This inspiration was simply the striking diagrammatic resemblances between chemical radicles and propositional logic:

A rhema is somewhat closely analogous to a chemical atom or radicle with unsaturated bonds. A non-relative rhema is like a univalent radicle; it has but one unsaturated bond. A relative rhema is like a multivalent radicle. The blanks of a rhema can only be filled by terms, or, what is the same thing, by “something which” (or the like) followed by a rhema; or, two can be filled together by means of “itself” or the like. (Peirce CP 3.421)

In the third section, we return to Peirce’s Beta Graphs when conceiving the identity line in Peircean iconic logic as the identification of a topological property, and we extend the importance of lines of identity. First, however, we look at some practical consequences of Peircean (non-)reduction thinking.

3 Biology and liberal naturalism

In the previous section, we attempted to illustrate the basic components of Peirce’s NonReduction thesis in the context of the Logic of Relatives and demonstrated how the idea of (non-)reduction is indissociable from diagrammatic logical representations (EGs). In the following section, we try to draw a potential scientific, namely practical, outcome from iconic logic and the NonReduction Theorem. In particular, we apply iconic logic to some key concepts in biology. We maintain that biology has inherent potential as an exemplar of iconic logic in that much current biological research such as evolutionary biology, proteomics, evo-devo, epigenetics, and other branches of biological explanations of a rather relational and topological type are manifest. ^[9] The standpoint of iconic logic also has both a logical character and a semiotic character, since the basis for his elaborated logic has evolved from primitive forms of semiosis (logic and semiotics are interrelated notions for Peirce).

The NonReduction Theorem at first sight appears to be in opposition to explanations in the natural sciences, which are mostly based on some kind of reduction, especially within naturalistic approaches. The core idea of explanations in modern sciences is viewing natural phenomena from a reductionist point of view with reductionism itself as a scientific standpoint enhanced in philosophical thinking since Quine and which is generally accepted by the scientific community.

Reductionism becomes controversial when applied to the social and human sciences. Nevertheless, reductionism might also become controversial in some particular areas of natural science, especially in biology. Proteins illustrate this. Proteins are biological macromolecules composed of a folded peptide chain and characterized mostly by their function, which cannot be simply and exhaustively explained by the chemical properties of a peptide chain (reductionism). Many peptide chains can acquire different functions depending on the actual topological context. These proteins are also called “topological” or, somewhat poetically, “geographical moonlight proteins” (Henderson and Martin 2011). Dupré (2010) asserts:

The important property of proteins in biological cells is their ability to interact with other molecules. An important subclass, enzymes, catalyse chemical changes to such othermolecules. But, first, being a particular kind of enzyme, a protease or a DNA polymerase, say, is a relational property among the enzyme, the substrate to which it bonds and the transformation that it catalyses. As it happens, a particular protein may have several kinds of enzymatic activity, and what it actually does depends on where it is in the cell and what else is in its vicinity [...] This example, finally, exemplifies a central general claim I want to make against reductionist positions: reductionist methods explain how it is possible for an entity to have a particular capacity, but to understand what capacities it exercises, and even, I want to say, what capacities it actually rather than merely possibly has, require seeing the entity in a larger context. (Dupré 2010: 292)

Dupré’s conception of proteins in general and “geographical moonlight” proteins (Henderson and Martin 2011; Jeffery 2014) in particular cannot be explained by a naturalistic-reductionist perspective, if for no other reason that their very functional identity is given by interaction with surroundings. In other words, a protein as a three-dimensional structure cannot be explained reductively by the chemical properties of the peptide chain (understood as a one-dimensional sequence) that constitutes it. This is because it acquires its essence (function) only when the peptide chain is folded and only when it enters into a relation with the surrounding molecules. By its definition, a peptide chain is merely a juxtaposition of amino acids or a sequence consisting of dyadic relations. A protein is a three-dimensional structure whose function is defined by this very structure. If we decompose the structure into a sequence, we formulate a reduction to dyadic relations, which means that we can no longer envisage the original structure, and consequently the function. ^[10] The relation between a protein’s function, a peptide chain, and the process of folding is a complex relation that, in our view, cannot be reduced to dyadic relations within the peptide chain. There is also empirical evidence about the problematic reduction of proteins into dyadic relations, and it is clear that, notwithstanding a huge area of research, to date we are unable to predict protein structure from its peptide chain (Backofen 2001).

As another example from the biological context, we take the relation between genotype and phenotype. The classical dyadic explanations comprehend phenotype as simple blueprints of genotypes, that is, they presuppose a dyadic unequivocal correspondence between genotype and phenotype. As an alternative, genotype-phenotype maps were proposed to explain the whole biological passage from digital script to a three-dimensional body more complexly (Huneman 2010).

The Modern Synthesis of evolutionary biology is a concept based on a one-to-one correspondence between genotype and phenotype that lacks further clarifications on how phenotypical variations are possible without a change in the DNA script, or vice versa, or how it is possible that a given phenotype may be obtained by different DNA scripts. The Modern Synthesis answer to the question of what the relation is between genes and their actual expression is that phenotypes are simply imprints of genes, this relation being understood as an unequivocal correspondence.

Recent studies in evolutionary biology, genomics, proteomics, and other life sciences expose the untenability of this elegant, but too simplistic model. Studies on evolvability and genotype → phenotype mapping (Alberch 1991; Pigliucci 2010) are conceived of as occupying the missing element – the missing transitional link between genes and their expressions. Studies on evolvability are to occupy the third position within the Peircean triad, the linking element not only between genotype and phenotype, but also between one genotype and another genotype, assuring connections between the whole genetic space, which is a form of synechism.

Another important consequence of understanding the correspondence between genotype and phenotype triadically is the dynamic component that Peircean interpretive semiotics brings. Interpretive semiotics is characterized by the unlimited endless semiosis, which means that the object of a sign signified by an interpretant might become in turn, an object of another sign ad infinitum. Semiosis is a dynamic process, so the correspondence between genotype and phenotype is, potentially, dynamic. The reason for introducing semiotics in this place is that the results of the extended synthesis disclaim purely dyadic explanations of elementary biological processes.

In contrast, a kind of space for interpretation (semiotic interpretation) is presented as an alternative. Eco (1997) had already assigned a semiotic nature to biological processes, naming it “primary iconism” or the “lower threshold of semiotics.”

As Massimo Pigliucci (2010) noted,

The undeniable progress we have made in understanding G → P maps, both empirically and theoretically, is such that one should hope that evolutionary biology has reached the point of forever being past simplistic ideas like genetic programmes and blueprints, embracing instead a more nuanced understanding of the complexity and variety of life. (Pigliucci 2010: 564)

Elsewhere Pigliucci explains the importance of the concept of genotype → phenotype mapping and the need to abandon the gene-centralized model:

Genomics and what I refer to as “postgenomics” (proteomics, metabolomics, etc.) started out squarely within the conceptual framework of the rather gene-centric MS, with the view that once we “decode” the genome of an organism we somehow gain a universal key to understanding its biology. The reality of organismal complexity has shattered such simplistic visions [...] The complexity of the genotype → phenotype map cannot be understood only by bottom-up approaches such as those that focus on gene networks and regulatory evolution, however. (Pigliucci 2009: 223)

In this section, we applied the basic features of iconic logic to recent studies in biology, specifically to protein studies and to the extended synthesis of evolution theory. Indeed, elementary molecular processes fit well into the approach of iconic logic because the very notion of iconism (primary iconism) was already applied to the molecular processes (Eco 1990, 1997).

With the evidence of aforementioned examples from biology, we would like to proceed in our argumentation to propose an alternative approach to a reductionist one within the philosophy of science. Dupré (2010) suggests that being a naturalist per se is unproblematic; only reductionism is quite contradictory with some elementary principles of naturalism. This is why Dupré proposes the standpoint of liberal naturalism. A number of scholars have elaborated liberal naturalism as a philosophical concept (De Caro 2010), but to date there has been no consensus on the definition of liberal naturalism.

The notion of the applicability of “topology” in biology was already introduced by Lindsely (2005). De Caro and Macarthur (2010) defined Liberal Naturalism as follows.

Liberal Naturalism [...] is best thought of as occupying the typically overlooked conceptual space between Scientific Naturalism and Supernaturalism. A necessary condition for a view’s being a version of Liberal Naturalism is that it rejects Scientific Naturalism, hence that it rejects the ontological doctrine or methodological doctrine, or both. (De Caro and Macarthur 2010: 9) ^[11]

As De Caro has stressed, the definition of liberal naturalism varies from author to author. Nevertheless, some basic features are shared by all approaches, such as the main features of liberal naturalism offering the possibility of a plurality of explanations, not only non-super-natural but also scientific. Ram Neta has formulated a dilemma for this:

What if digestion, or respiration, or reasoning are natural kinds, their nature consisting simply in the mechanisms enable them to occur? Is the liberal naturalist committed to denying this possibility? If so, then I confess I can see no good reason to accept Liberal Naturalism. And if not, then I do not understand just what Liberal Naturalism is. (Neta 2007, as cited in De Caro and Macarthur 2010: 70)

We refute the subsumption of liberal naturalism under scientific naturalism and supernaturalism. We hold instead that scientific naturalism may be re-established, as Dupré proposes. Our liberal naturalism is, thus, liberal in two aspects: it is sensitive to normativity, and it replaces mechanistic explanation with a new form of explanation – a topological explanation.

Neta’s understanding of liberal naturalism is a dogmatic way of understanding this philosophical concept. This is a paradox if we take into account that liberal naturalism is striving to deliberate from a dogmatism imposed by classical naturalism, as for instance Williamson has pointed out: “Naturalism as a dogma is one more enemy of the scientific spirit” (Williamson 2014: 31).

We hold that for naturalism to be considered non-dogmatic it has to fulfill these conditions:

1. Normativity is considered only as a first principle (see Williamson 2014: 37) and does not interrupt the structure of explanation – i.e. it is not the return of teleology. As Williamson says:

   If it is true that all truths are discoverable by hard science, then it is discoverable by hard science that all truths are discoverable by hard science. But it is not discoverable by hard science that all truths are discoverable by hard science. Therefore, the extreme naturalist claim is not true. ‘Are all truths discoverable by hard science?’ is not itself a question of hard science. (Williamson 2014: 37)

2. The power of naturalism has to be preserved, which is expressed in the need for empirical adequacy of scientific models, unification of theories, and comprehensibility of the world (see van Fraassen’s “Empirical Stance” 2002: 41–46 and Williamson 2014: 31).

3. The prevailing type of mechanistic explanation has to be augmented by a non-mechanistic type of explanation, which in our case is the topological model of explanation. In this. the assumption of reducibility must be rejected. ^[12] Ontologicalproviso:

   There may be entities that do not and cannot causally affect the world investigated by sciences and that are both irreducible to and ontologically independent of entities accountable by science but are not supernatural either, since they do not and cannot violate any laws of nature.” (De Caro and Macarthur 2010: 75–76)

   1. Non-reducibility does not mean, however, abandoning the criteria expressed in point 2, because of the possibility of making use of iconic logic in connection with topological explanations. More precisely, iconic logic would provide the same degree of formal adequacy as symbolic logic without falling into the pitfalls we have previously mentioned.

We call a naturalism that fulfills conditions 1 to 3.1 liberal.

Liberal Naturalism seems to constitute a useful way of understanding natural phenomena as natural, that is non-teleological, yet not entirely reduced to mechanistic explanations. A universal trend in the philosophy of science recently has been to consider explanations different from mechanistic ones (e.g. Reutlinger 2016).

Natural species are not strictly definable by means of mechanistic explanations, and as a demonstration the NonReduction Theorem can be applied. In our understanding, liberal naturalism is simply naturalism without mechanism, thus without reductionism, as proposed by Dupré. In this way, liberal naturalism is vindicated as a scientific approach.

We maintain that the Ram Neta dilemma for liberal naturalism can be solved with our stated position. Our liberal naturalism is still scientific naturalism which rejects supra-naturalism, but concedes a liberal space for diverse kinds of explanations. The liberal naturalist holds that, if we recall Ram Neta’s example, “digestion, respiration, reasoning,etc.” are accessible to scientific exploration. The term “mechanisms” in the Ram Neta dilemma is too dogmatic, however, in the context of an explanation. Naturalism bereft of a strictly mechanistic mode of explanation is the sine qua non of liberal naturalism.

We have defined the NonReduction Theorem in the first section and liberal naturalism in the second section. We now proceed to the first tentative conclusion of this paper, to propose a nexus between iconic logic and liberal naturalism.

4 Mechanistic vs. topological explanations

In the previous section, we proposed conceiving liberal naturalism merely as a naturalistic source of explanation dissociated from reductionism and mechanism (continuing in Dupré’s argumentation). This means, however, that mechanism has to be replaced by another means of explanation. We propose replacing mechanistic explanations with topological explanations.

The notion of topological explanation is succinctly expressed in Huneman (2015):

Topological explanation is an explanation in which a feature, a trait, a property or an outcome X of a system S is explained by the fact that it possesses specific topological properties Ti. (Huneman 2015: 117–119) ^[13]

Whenever the explanandum—a property, outcome, behavior of S—is explained by the fact that the system has topological properties Ti, I thereby say that a topological explanation has been given. “Explanations” here means that some fact G is entailed by the topological properties Ti, and is itself a mathematical fact that describes adequately the explanandum under focus. (Huneman 2015: 117–119)

Where Huneman uses topological graph theory and network theory as a basis for his model of explanation, we try to establish a formal logical core of a model topological explanation. We assert that the core lies in applying some kind of iconic logic. In our case, it is the form of iconic logic, as delineated above.

The topological explanation is widely used today in conflict with a mechanistic explanation (Huneman 2015, 2010; Kostic 2018), especially in the context of biology. This term within the philosophy of science was thoroughly defined by Huneman (2015, 2010), referring to the non-metaphorical and non-analogical use of the concept of topological properties as an invariant of the continuous transformation of topological space in topology and the topological theory of graphs (see Gross and Tucker 1987). Nevertheless, the authors who favor this concept use it mostly as a mere “explanatory sketch,” that is firmly based on a specific example (again, Huneman 2015, 2010; Kostic 2016). There is therefore no guarantee that the same sketch will also apply to other cases. In other words, the explanatory power of the topological explanation is attenuated because of a lack of universal applicability.

We assert that iconic logic and the notion of the iconic (diagrammatic) sign represents one of the roots of topology and the topological theory of graphs. Previous studies focused primarily on the introduction of some mathematical and topological concepts by which Peirce has been compared with Poincaré (Kauffman 2001; Caterina and Rocco 2016; but mainly Havenel 2010). We also hold that the semiotic approach, in this case based on an icon (iconic sign) in a diagrammatic sense, provides a conceptual tool based on the Logic of Relatives and EGs, which can be used to interpret the topological explanation in a universal form, not only related to specific examples.

In Peirce’s EG, logical and topological considerations intertwine (see also Zalamea 2017). Triads are simply topological entities with their valency properties corresponding to a certain class of topological graphs (Gross and Tucker 1987). We can assert that the RCT theorem is basically a topological theorem. There are also other topological expressions for EG (Havenel 2010). The claim that iconic logic is formal logic can be upheld as more appropriate when iconic means topological.

It is crucial to remember that by iconicity we mean the specific case of diagrammatic iconicity, which is characterized by the resemblance of the internal structure, or the resemblance of relations between two objects. This is different from imagic iconicity, based exclusively on visual resemblance. In this manner, an identity line in Beta Graphs is a perfect example of diagrammatic iconicity. The identity line, as an iconic sign of closeness, resembles the intimacy of two concepts (e.g. two syntactic objects). It resembles this intimacy, however, not visually but relationally and thus diagrammatically or topologically.

If we state that we can interpret the Logic of Relatives and teridentity topologically, an important question arises: Where and what represents the basic topological notion at the heart of the iconic logic? Obviously, we can say that topology is closer to diagrammatic/iconic logic precisely because it is important to preserve certain structural invariants (i.e. topological properties). ^[14] In EG (Beta Graphs in particular), it is again the structure of the graph, i.e. the invariance of the given property, and thus the identity line can be understood as a topological entity sui generis. Whenever a placement is involved, iconicity is introduced in the sense of representing certain invariant spatial relations, and that again signifies topology. ^[15]

We maintain that the most important topological property in EG is expressed by identity lines. The extra-information contained in the EG consists in the determination of the invariant (topological), i.e. the detection of a topological property. Iconicity is a certain structural/topological attribute that is presented in Beta Graphs by Identity Lines. To determine identity therefore means to find a topological property, not to find a similarity in a broad sense. ^[16] We hold that the term “topological” is close to the term “semiotic” in this respect. ^[17]

Thus, in the current tendency toward topological explanations in biology, more than just a new kind of description of performing a given scientific discipline can be proposed. It can be a real transformation of the concept of biology toward mathematics. If concepts such as the NonReduction Theorem and teridentity are applied in biology, mechanistic explanations can be replaced by topological/semiotic explanations.

We equate topology and semiotics only in a narrow context which is related to the Peircean mathematical definition of topology and continuum applied to his own semiotic theory. We are aware that semiotics in the broad sense is not reducible to topology, yet semiotics understood rather as an adjunct of logic, as proposed by Peirce, might be comparable with mathematical topology. This might allow in this manner for one more variation of non-mechanistic explanations in the philosophy of science.

5 Conclusions

The aim of this paper was to demonstrate on a theoretical level (philosophy of science) and on an applied level (current research in biology) that iconic logic and the NonReduction Theorem are pivotal concepts. This applies not only from the point of view of the history of ideas but also because they are still articulate in the current scientific paradigm. We analyzed the nature of Peirce’s works in logic and how they might be rearticulated with an optic of topological explanations, this being one of the fruitful fields of the contemporary philosophy of science. We identified two possible applications of the Peircean triadic relation to biology, protein folding and genotype-phenotype maps. We also proposed potential varieties of non-mechanistic explanations in the context of biology, focusing on the philosophy of Charles S. Peirce and his semiotic-logical works on (non-)reduction. Our motives were oriented on the practical viewpoint, and from here Peirce’s logical works, notably his iconic logic (EG), are better suited to express biological phenomena when comprehended in a non-mechanistic sense. The moment we decide to amend mechanistic explanations in biology, we have eo ipso to deal with a plethora of biological processes that are difficult to describe through the medium of a linear script. Genotype-phenotype maps are proof of the difficulty of describing such complex and irreducible processes to a linear transcription.

Summa summarum: determining an identity means discovering a topological property. In Beta Graphs, the line of identity is an invariant of continuous transformations of a given system, i.e. it represents a topological property. In this way Beta Graphs should constitute an appropriate logical basis for topological explanations.