Peirce's incomplete synthetic turn.
The project is not the proof itself. Rather, it is the reason for which the proof could not attain a final form. The proof never satisfied Peirce because it was always a partial view of something broader that Peirce usually identified as being "the truth of continuity" or "the truth of synechism," (3) or else the method of justifying the "nature of Sequence," (4) namely, the nature of pragmatism. However, the proof required a vast panoptic view necessarily covering different fields. In a letter to William James, he wrote that even if every single part of his work had been published, "the principal thing would remain unpublished; for this depends upon the way the parts are filled together." (5) This comment raises the question: what was the overall project?
This paper intends to answer this question and possibly to complete it. The first part of the article (sections I and II) identifies in a "completely" synthetic pattern of reasoning, the direction toward which Peirce was leaning in his research. It also underlines that Peirce failed to provide a satisfactory account of this syntheticity. He forged many new "synthetic tools," but he did not put them in a new pattern, and he did not change the definition of "syntheticity."
The second part (III-IV) explicates a Peirce-driven definition of "synthesis" compatible with Peirce's insights taken from his mathematical and logical studies.
The third part (V) proposes a "completely" synthetic theory of knowledge based on the new concept of "synthesis" elaborated in the previous part. "Complete gestures" as meaningful actions that embody universals in particulars are the basic elements of this account of the dynamic of knowledge.
It was in the "Illustrations of the logic of science" (1877-78) that Peirce had first furnished his own version of the logic of inquiry, the core of which was the pragmatic maxim within the realistic procedure of science. Notwithstanding the clarity of the maxim, his logic of inquiry presented also several difficulties that Peirce tried to fix by deepening and extending his logic, which required the elaboration, at the turn of the century, of an entire classification of sciences and the study of several interdisciplinary topics. (6) In three series of lectures Peirce gave from 1898 to 1903, he tried several possible orders for these scattered parts of his philosophy. An ordered list of topics could be the one proposed in "Prolegomena to an Apology for Pragmaticism": continuity, phaneroscopy, signs, existential graphs, kinds of reasoning. In the series written for The Monist (1905-6) he stressed also the role of normative sciences. In the following years he tried in vain to fully explain this or a similar order he had in mind.
These efforts were "in vain" because Peirce tended to lose his track while tilling the "virgin soil" (7) of those many fields of research he himself discovered.
Here a complementary question arises: why did he get lost?
Part of the reason is that all the above topics were products of his original insight, and Peirce was eager to explain them precisely. The mathematical definition of continuity stemmed from his thirty years of studying Cantor's set theory, and Peirce had independently discovered Cantor's theorem (the cardinality, or the multitude, of the set of all subsets of any set is strictly greater than the cardinality of the set) and Cantor's subsequent paradox (the set of all sets is at once larger and smaller than the set of its subsets). In contrast to the German mathematician, Peirce saw the immense philosophical impact of this discovery. In his lectures and in his papers or drafts he spent a lot of time explaining the mathematical basis of the theorem and of the paradox, saving little time and space for their philosophical impact. Peirce perhaps was more successful with his explanation in phaneroscopy, semiotic, and logic. In these three fields he really succeeded in accounting for the importance of those discoveries to our philosophical understanding and their impact on this understanding. Nonetheless, Peirce failed to provide a definitive sketch of semiotic grammar, (8) a thorough account of abduction, and the actual description of the way in which we carry on a phaneroscopic analysis. (9)
Again, all these topics were brand new fields of inquiry and Peirce had to spend most of his time convincing his reader or audience (or himself) that these fields existed and were worth studying. What deserves attention is his statement that the real explanation of all the new topics lay in their connection to one another (10) and in the proof this connection affords for pragmatism, that is, for the sequence of thought. That what he meant by the statement is not entirely clear is no objection but rather more fuel for further inquiry.
Another part of the reason for Peirce's failure in giving a final account of pragmatism is that he was looking for a completely new pattern of reasoning, even though he was not completely aware that he was doing so. Which was the completely different pattern? We can call it a "completely" synthetic pattern. What does it mean?
The distinction synthetic/analytic is one of the most powerful of the history of philosophy, and the discussion led by Kant established a canon for such a topic. Peirce did not discuss this distinction directly, and probably he accepted more from the German master than he was aware of, but it is certainly significant that he early argued that the real question did not concern "How are a priori synthetic judgments possible" but "how are synthetic judgments possible at all?" (11) Notably, this is the beginning of those refusals of intuition and of transcendentalism that led him to his semiotic pattern. However, here I want to discuss this plea for syntheticity by following another track established in Peirce's Existential Graphs (EGs hereafter), the iconic logic that he thought of as his "chef d'oeuvre" and that was meant to represent as closely as possible our actual reasoning in both everyday life and scientific inquiry.
What is synthesis? As we know from Kant, synthesis means to amplify a judgment by something altogether different from the concept given in the subject. (12) This "something" can be drawn from pure intuition (a priori synthetic judgment) or ultimately from sense perception (a posteriori synthetic judgment). As we know, one of the difficult and most notable achievements of Kant's understanding is to explain how mathematics can be "synthetic" even though this syntheticity has to be justified a priori.
Peirce seemed to understand this syntheticity profoundly, (13) and, thanks to his mathematical studies on continuity, he did not think he had to rely upon any kind of a-prioricity. We could say that he followed Kant's synthetic intention more than Kant himself. Peirce's synthetic reading resulted in an original view of the relationship between logic and mathematics, very different from the logicist one. Eventually Peirce founded logic upon mathematics, where mathematics was always meant to be the "synthetic" study of universals in particulars. In mathematics, as we will see, the use of particulars is the use of diagrams, and Peirce tried to formulate an iconic logic--the EGs--which has the same property. Therefore, EGs prove to be Peirce's attempt to found a "synthetic logic".
Peirce reached good results with his EGs. Moreover, he signaled in them the necessity of representing modalities and change as essential elements of our usual thought. This was a new aspect of his thought, and it is a crucial part of his attempt to link his logical-mathematical studies on continuity with his analyses in EGs. However, Peirce did not succeed in giving a complete account of our knowledge according to this synthetic point of view because he failed in putting all his synthetic tools together in a synthetic theory of inquiry or knowledge. EGs remained an analysis of the synthesis of the dynamic of our actual reasoning. It is this conjunction of analysis and change that creates a tension which Peirce tried to work out till the end of his life.
In my view, the problem is that the Kantian view of syntheticity with which Peirce stuck could not express the far greater advancement to which Peirce's philosophy itself aimed. There were at least three points on which his philosophy exceeded the Kantian comprehension: i) his logic comprehended the genesis of concepts from perception through signs in a single a posteriori pattern; ii) his view of poliadiac logic, induction, abduction, and modalities put his studies on reasoning way beyond Kant's understanding; iii) his use of "individuals" as diagrams for analyzing thought was much more synthetic than any analyses dreamt of by his predecessors.
Peirce did not have the strength to pull the ties together, and in my opinion, the paradigm never became a "completely" synthetic pattern in classic pragmatism notwithstanding Dewey's and Mead's attempts. (14)
Therefore, I will try now to give a different pattern for analysis and synthesis that would account for the mentioned elements, and then I will use Peirce's studies for explaining this paradigm as far as possible. In the final part, I will give a final synthetic turn with a personal suggestion on the method through which this synthetic reasoning happens in our everyday life.
This proposal of a new paradigm of reasoning would respect Peirce's discoveries and, in my opinion, would also help contemporary philosophy toward a real antidualist view of reasoning more respectful of our everyday reasoning. In the following discussion, three definitions that I propose here should be kept in mind.
A synthetic judgment (and reasoning) is a judgment (and reasoning) that recognizes identity through changes.
An analytical judgment (and reasoning) is a judgment (and reasoning) that loses identity through changes.
A vague judgment (and reasoning) is a judgment (and reasoning) that it is blind to identity through changes.
The first task in order to show the plausibility of our paradigm is to explain what change is and how we can study it. The two questions go together if we use Peirce's fundamental insight: the concept of continuity, the "keystone" of pragmatism. In order to understand change in this sense, we need to use a mathematical and a logical approach.
From the very beginning of Peirce's intellectual proposal, continuity played a central role. Continuity was paired with representation, or cognition, already in the 1868-69 Journal of Speculative Philosophy series, and this pairing became increasingly unified the more he was discovering the mathematical structure of continuity. (15)
There are many changes in Peirce's mathematical approach to the topic, but substantially they all focus on the proof of Cantor's theorem and paradox that Peirce independently discovered in the late 1890s. With this proof, Peirce understood that there is an infinite series of multitudes that Cantor's set theory can reach, but that those multitudes are always bounded to an imperfect or pseudocontinuity that depends on the unavoidable singularity of the initial definition of set or collection. (16) Peirce's definition of a set or collection implies that a collection is an individual whose existence depends on the regularities among other individuals. These "regularities" can be identified by the characters its members possess (ineunts) or exclude (exeunts). To be singularly existent means to have one quality that defines the collection while the rest of the universe does not have that quality or does not have it in the same sense. (17)
Therefore, the definition of collection leads immediately to the impossibility of grasping the "totality" by increasing multitudes. If a collection implies by definition ineunts and exeunts, namely, a scheme of otherness, the collection of all the collections, not having by definition any exeunt, is unthinkable. Cantor's paradox mathematically confirms this evidence that Peirce first attains through the categorical status of individual. Therefore, the real continuity is beyond any calculation that set theory can reach.
What is this "real" or "perfect" continuity, which is beyond the "pseudocontinuity" that sets can reach?
Peirce changed his mind many times on this issue, trying first to tie continuity to necessity and then to possibility. (18) Peirce first thought of making continuity the complete evolution of reality, the perfect "generality" in his logical terms. In this version (1897-1905), any singularity is a rupture of perfect continuity like a chalk mark breaks the continuity of a blackboard. (19) Afterwards (1907-1914), he connected continuity to a more complex pattern in which continuity is a possibility, namely, a model that may be realized. Singularities are now realizations of that original possibility tending to a general cohesiveness. (20)
It is in this second sense that our conception of change must be seen. We can understand change as a perfect continuity of possibilities of which any actual occurrence is a realization. In other words, whenever we reason about something existent we are also reasoning about something that changes according to a possible model and following general conditions. Using Peirce's insight about continuity, we understand this change not as a property but as a reality to which our existent things belong. So if we want to explain this "belonging," we have really to refer to what continuity is. As we will see later, Peirce would add modalities to his explanation in order to specify what he meant. However, for now, let us stick with Peirce's definition according to which continuity is a law (general) whose internal regularity is "an immediate connection" that we can understand as the condition of every possible realization. (21)
According to Zalamea, we can define a Peircean perfect continuum by four characters: modality (plasticity), transitivity, generality, and reflexivity, each underlying one aspect of the relationship between the parts and the whole of continuity. (22) I will take Zalamea's terms in their philosophical meaning. "Generality" is the law of cohesiveness among parts beyond any individual and any possibility of metrically measuring it; "modality" means plasticity, namely the fact that a continuum is not tied to actualities but involves both possibility and necessity; "transitivity" is the internal passage between modalities (possibility, actuality, and general necessity); "reflexivity" means that any part shall have the same properties of the whole to which it belongs.
How should we study this "real or perfect" continuity? Peirce's answer is: by EGs. I will try to exploit this insight later on. In preparation for it, I want to underline a characteristic of Peirce's studies on mathematics that will be decisive for both EGs and our logical paradigm.
The discovery of Cantor's paradox did not generate in Peirce any doubt about the foundations of mathematics. How is that possible? The answer is that perfect continuity is real, and as such it guarantees our reasoning even though the foundation reality can provide will come a posteriori and not a priori. How could Peirce be so sure about it? It is part of the maxim of pragmatism--and indeed of our everyday synthetic reasoning--that "working" is the necessary and sufficient condition of reality. "If it works, it is." According to this maxim, our "doing mathematics," our scribing graphs and diagrams whether on the sheet of the mind or on some physical sheet, works: mathematics is one of the most developed and successful sciences, and according to Peirce it is so because it imagines hypotheses and draws from them necessary conclusions. Therefore, if we cannot doubt that it works, at the most we can wonder how it works.
Here we find the first hint of the "syntheticity" for which we are looking. Mathematical diagrams work because they act synthetically. According to an old Kantian definition that Peirce knew very well, in mathematics we are dealing with universals in particulars, while in philosophy we have to deal with universals abstracted from particulars. (23) The great power of generalization of mathematics is due to these contracted universals: our image of a triangle is good for any triangle and for any of its properties because in the particular triangle we draw on the blackboard we are contracting the entire universal "triangle." Therefore, doing mathematics means already dealing with the reality of universals, and there is no surprise in being sure that while we are doing mathematics, we are participating in the construction of a broader metaphysical reality.
I will call "mathematical gesture" this kind of synthetic approach to mathematics through our "doing mathematics."
There is a second interrelated approach to "change" in Peirce's late writings. It is the approach through logical modalities. Peirce's understanding of logical modalities grew during the years, reaching a more and more realistic view. Possibility, actuality, and necessity became the ways in which Peirce explained transition within the continuum itself. If Cantor's paradox showed the existence of an original higher continuum of reality that exceeds our computations, modalities define the internal life of this continuity. As we have already seen, Peirce's understanding of the modality of continuity swerved from necessity to possibility and found a final version in which continuity is made of possibilities whose internal cohesiveness is necessary. In what do these modalities consist?
Summing up Peirce's conception, possibility is the mode of reality in which the principle of contradiction does not hold, namely two alternative options can be true at the same time and according to the same aspect. Actuality or existence is the mode of reality in which both the principle of contradiction and the excluded third hold. Necessity is the mode of reality in which the principle of excluded third does not hold, that is, both alternatives can be false. Peirce's short version of the discovered ontological modalities sums them up as "may be's" (possibility), "actualities," and "would be's" (necessity or generality).
These ontological definitions imply that all three modalities are "real"--meaning that they are independent from what any number of minds can think--or rather they coincide with the evolving continuity of reality. Reality actualizes possibilities and tries to develop them as generalities, in an ongoing process of growing reasonableness. (24) According to Peirce, it is impossible to describe reality without modalities.
The same internal life of the modalities is replicated within our lower-level logic in conformity with the property of reflexivity. The logic of modalities implies a different understanding of stechiology (doctrine of elements). A term can be vague, determinate, and general according to the characters--both positively and negatively--inhering in them. (25) If any character is predicated universally and affirmatively or recognized as inherent, the term is determinate. If it requires further determination by the utterer, it is vague; if it requires further determination by the interpretant, it is general. Vagueness, determination, and generality fall under the same description of the logical modalities, and they manifest the impact of logical modalities on the elements of our everyday assertions.
What we have here is a transition in determination which is based on the ontological reality of the logical modalities.
In order to show how this transition occurs, I will stress the importance Peirce ascribes to the first part of this trichotomy. Vagueness has such an importance because it is the possible state of things from which ideas stem: our ideas start as vague because they are a not yet determinate part of reality. In this vagueness further rich determination is implied as possibility. Our vague judgments can be the spring of future actualities and necessities. Scientific discoveries, existential beliefs, and even personal relationships always begin in a vague hypothetical state that is the humus for any further fruitful determination. (26)
Let us now come to the definition of change that we can state after the considerations on Peirce's philosophy that we have done so far. Our stance for realism allows this definition to hold both for "change" and "changing something": a continuous reality in continuous transition among modalities. Logical features respect Zalamea's four characteristics of the continuum: "reflexivity," "generality," "plasticity," and "transitivity." Every element has the same properties of general modalities (reflexivity); the passage among vagueness, determination, and generality (transitivity) is the law of the development of meaning through categories (generality); this latter depends on the possible inchoative status of vague meaning (plasticity).
How are we going to take it into account in our reasoning? We said that synthetic reasoning is about "recognizing an identity through changes" and that the other kinds of reasoning derive their definition from this one. Now, knowing a little more what change is in a Peirce-derived conception, we need to understand what we mean by "recognizing."
In the history of philosophy, there are at least two principal models for recognition: one provides for the permanence of certain attributes that allow us to identify an object, the other refers somehow to the Hegelian dialectic, which then takes place in an idealistic or hermeneutic sphere. (27)
Neither the recognition of attributes nor dialectic recognition seem to have the characteristics we need: we want to trace the logical pattern of recognizing, and, at the same time, avoid the presupposition of fixed attributes. It is another case of the application of the pragmatist "third way" to philosophical issues: a way that preserves both a precise method and an open interpretation.
In order to understand what is going on in our "synthetic reasoning," the reasoning mostly used in everyday life and in many difficult cases of hypothetical reasoning (scientific discoveries, medical diagnoses, trials based on circumstantial evidences) or pragmatic understanding of meaning, we can rely on Peirce's study of the EGs, which were the way in which Peirce himself sought to represent continuity, and they are possibly one of the few scientific strategies of getting to the structure of "recognizing." As a matter of fact, EGs are an alternative way to logic: a "map of thought" not by symbolization but by iconization. (28)
As many authors have pointed out, EGs are the iconic formalization of logic of propositions (Alpha), first order (Beta), and modalities (Gamma). (29)
Sure enough, EGs are still analytic insofar as they draw necessary inferences, but, as already stated, their overall (almost unconscious) project is a synthetic one. Insofar as they are a "map," (30) or "a moving picture of thought," (31) capable of representing the creation of explanatory conjectures, EGs try to grant the traceability of "critical common sense" and its evidence. Using the phrase introduced above, they are a kind of "mathematical gesture," and from this point of view, they are synthetically conveying universals into particulars. It is from "evidence" that we need to start in order to understand how EGs can be used to explain what "recognizing" is.
According to Peirce, EGs--conveying universals into particulars--can show the path of any reasoning with evidence and adequate generalization. (32)
Of course, at first reading, "evidence" does not sound like a very Peircean word, but the knowledge of the whole range of his research reveals that the mature Peirce was defending perception as a perception of generality. (33) So he thought that the diagram-icons enhanced by symbolic interpretation furnished by EGs could make us perceive the generality according to which a conclusion follows from premises. (34)
Clearly they cannot do it alone. In the same text, "Prolegomena for an Apology to Pragmatism," (35) Peirce explains that diagrammatic-icons could not give evidence if they did not involve a Symbol of which they are the Interpretants. Peirce explains through his analytic semiotic theory the way in which we pass from a diagram (transformand diagram) to another (transformate diagram) according to a scheme (the Kantian word is used to signify the combination of diagrammatic-icon and Initial Symbolic Interpretant) thanks to which we understand that the second diagram is somehow involved in the first. (36) Setting aside the analytic semiotic explanation, what we observe in diagrams is a generality. As we have seen, universals live in particulars, and even in singulars.
Besides evidence, we have here a second important character, which belongs to any mathematical analysis: mathematics has an enormous power of generalization. (37) If we analyze this property, we will find that it is due to what Peirce calls a "hypostatic abstraction," the logical tool through which we pass "from 'good' to 'goodness' and the like," (38) and we consider this abstraction as an object. (39) Accordingly, the abstraction that results from the diagrams both in mathematics and in EGs provides this power of generalization. How does it become real? Generalization happens in "doing" or "scribing" those diagrams. If the generalization is the analytical result of the diagrams, diagrams are the synthetic happening of generals.
"Recognizing" is part of this synthetic "happening" and should have the same characters: evidence and generalization.
Recognizing always implies an object: recognizing something; and the easiest way to look at it is to look at the case of identity. "Recognizing something" means at least to acknowledge an identity as far as semiotics and critical logic are concerned. Here we have to remember that the identity we are looking for is not a plain correspondence A=A, which according to Peirce himself is only a degenerate form of the real identity A=B. (40) Identities are always passing through changes. A=A is the static correspondence drawn from the set-theoretical definition of multitudes and it is a simplification of a "more primitive" form of relation. (41)
Given our diagrams with their power of evidence and generalization, how can they help us understand our "recognizing an identity"? We know that this recognizing has to happen within a continuous changing pattern. How could we represent this complex unity of change and identity?
Let us begin with the representation of continuity. The first kind of continuity, the continuity of reality, is represented by the sheet of assertion. That is the first continuity, easily stated in the Alpha and Beta parts, namely in the logic of propositions and predicate logic. The sheet represents the universe of discourse. (42)
However, the game gets more complicated when in the Gamma graphs we want to represent our everyday reasoning composed of changes and possibilities. This is why Peirce looks for a different kind of continuum. Not a sheet of assertion anymore, but a plastic multidimensional continuum.
You may regard the ordinary blank sheet of assertion as a film upon which there is, as it were, an undeveloped photograph of the facts in the universe.... But I ask you to imagine all the true propositions to have been formulated; and since facts blend into one another, it can only be in a continuum that we can conceive this to be done. This continuum must clearly have more dimensions than a surface or even than a solid. (43)
The multidimensional continuum is more "original," and the sheet of assertion of Alpha and Beta parts is only a picture of this original continuum. (44) In this continuum we are really scribing not the propositions as they actually are, but as they might be. That is why we could think of them as marks or qualities, which cannot be numerable anymore--in so far as they are possible, that is they refer to the continuity that surpasses the grasping power of the sets--and they will exceed any multitude, according to our definition of "perfect continuity." On a multidimensional continuum--greater than a solid--we are not writing "assertions" anymore. This multidimensional continuum is apt to represent time or "Becoming." (45)
Now that we have the tool for representing the continuum of time, namely of change, we have to look for identity. In the Beta Graphs we have a graphical tool for identity: the line of identity that can solve the problem of quantifiers necessary in a predicate logic. Peirce defines the line of identity as follows:
A heavily marked line without any sort of interruption (though its extremity may coincide with a point otherwise marked) shall, under the name of a line of identity, be a graph, subject to all the conventions relating to graphs, and asserting precisely the identity of the individuals denoted by its extremities. (46)
The line represents the existential quantifier when it is evenly enclosed and the universal quantifier when it is oddly enclosed. (47) Once again, the definition becomes more complex when we have to deal with the multidimensional plastic continuum.
When we scribe a line of identity on the multidimensional plastic continuum we are representing the identity into the continuum of possibilities. Identity continues to assert the identity of the individuals denoted by its extremities, but of course the kind of identity we are drawing becomes very different.
To say that who commanded the French in the battle of Leipzig commanded them in the final battle of Waterloo, is not merely a statement of identity: it is a statement of Becoming. There is an existential continuity in time between the two events. But so understood the statement asserts no Significative Identity, inasmuch as the intervening continuum is a continuum of Assertoric Truths. Now, upon a continuous line there are no points (where the line is continuous). There is only room for points--possibilities of points. Yet it is through that continuum, that line of generalization of possibilities, that the actual point at one extremity necessarily leads to the actual point at the other extremity. (48)
Identity here means the continuity of possibilities of an individual considered as a changing object in its becoming. Identity is not anymore A=A, but a nonpurely-symbolizable iconic identity passing from A to B.
This switch also implies that the line of identity when it is scribed upon a multidimensional continuum must become a line of teridentity, namely a line that represents two relations of coidentity, (49) in which one of the extremities is a loose end, that is, an extremity of a line of identity not abutting upon another such line in another area. Let us illustrate the difference among different representations of the line of identity as drawn by Peirce.
Line of identity abutting upon the same area:
"Some black bird is thievish" (50)
Line of identity abutting upon two different areas:
"There is something which, if it be a salamander, lives in fire." (51)
Teridentity means that there are two different relations of identity that have in common one end and part of the line. As Peirce puts it, "it is identity and identity, but this 'and' is a distinct concept, and it is precisely that of teridentity." (53) We need a loose end in order to guarantee the realization of a different possibility. Napoleon that won at Leipzig is the same man who lost at Waterloo, but he might have been also the man who won at Waterloo, if the mud had not prevented his military maneuver. In this case, the loose end differs slightly from the actual end, but the line would depart earlier if the man who won at Leipzig decided not to invade Russia. (54)
Identity on a potential continuum has to present itself as the character of potentiality, otherwise it would lose the capacity of representing different possible ends. A loose end means that we know identity as a continuity but we do not know at which point it would stop and the point might be or have been different from the one we have. We have to allow for the possibility that brute existence will be different since we are not representing actuality but potentiality. Finally, the line of identity has a direction or an aim because we draw and read it starting from one point and ceasing at another. (55)
This is the tool Peirce crafted to define analytically the kind of identity we are dealing with in our different types of reasoning. However, in order to grasp the synthetic view of identity, it is not EGs' logical analytic use that is needed. The interesting features of the line of identity and teridentity come from Peirce's sporadic comments on the nature and the meaning of it.
The first comment is that the line of identity, and afortiori the line of teridentity, is another "perfect continuum" along with the multidimensional continuum of assertion. What does it mean?
Identity means continuity, not necessarily in Place, nor in Date, but in what I may call aspect, i.e. a variety of presentation and representation. (56)
The line of identity is the continuity of any presentation and representation of aspect of an individual. (57) Time and space are just two of the possibilities of these representations, even though probably the most important ones. If identity is a continuum, how can it respect the characters of reflexivity, generality, plasticity, and transitivity?
Before trying a definite answer, examining the elements that compose this line of identity will help us trace the physiognomy of the continuity of the line of identity.
What is the line of identity from a semiotic point of view? According to Peirce, the line is a "perfect sign," namely it blends almost equally all the signs. It is a general (a law), but it identifies individuals (index) in an iconic way (icon). (58) Every aspect of it is important: the symbolic level allows any replica of the line to be a general interpretation grasping individuals; the indexical level marks it starting from one finite point and ending up at another one; while the iconic level makes the line appear as "nothing but a continuum of dots, and the fact of the identity of a thing, seen under two aspects, consists merely in the continuity of being in passing from one apparition to another." (59)
The iconic level is the more important one because, according to Peirce, it shows "the Forms of the synthesis of the elements of thought." (60) Signs always represent an object by selecting a certain aspect in its abstract form. (61) Icons, working by similarity, represent this form in a mental or physical diagram. They bring the first form and the first feeling that an object "emanates." (62) In icons we can see the form of the relations that an object might have, if it were realized according to certain conditions. That is why to those Forms we owe mathematics' synthetic capacity with its huge power of evidence and generalization: they manifest the possibility of any relation that the object may have and would have, if realized. (63) Icons have the nature of "an appearance, and as such, strictly speaking, exist only in consciousness" (64) regardless of actual existence. They "merely suggest the possibility of that which they represent, being percepts minus the percussivity of percepts." (65) The icons that furnish form and feeling to the line of identity, metaphysically speaking, are possibilities.
We are now ready to verify our four characteristics of continuity. The line of identity is made of icons interpreted, and thus far is made of possibilities whose realizations are connected by a general rule. It is exactly like the "perfect continuum" described above according to the properties of plasticity and generality. The loose end guarantees that possibility would be open even if the two brutally cut ends seem to impose a certain kind of actuality. It also guarantees the fullness of reflexivity: any parts of the line, including the ends, are possibilities that might be realized according to a general law. The perfect continuum of the line of (ter)identity is written upon the perfect multidimensional continuum and every part of the two of them shares the same nature.
As for transitivity, every single dot actualized on the multidimensional continuum realizes a possibility according to the general law, and might have a different branch--the loose end--as a different actualization of the same possibility. So there is a passage from possibility to necessity through actuality, and even a passage from necessity to actuality. The translation of modalities in terms of vagueness, determination, and generalization is easy to see: identity is a progressive determination tending toward a generalization (tending to identify forever the winner of Leipzig and the loser of Waterloo).
Those technical passages help us to understand how Peirce thought that EGs could express both identity and recognition of identity. Identity is neither a dialectic nor a permanence of characters. Identity is a possible continuity of possible qualities ("aspects") that become more and more determinate during the development (the "becoming") expressed by the continuum within which they are inscribed. In other words, identity is a revisable development of the (dynamic) object through changes of time and space, or whatever aspect of the continuum we could imagine; or a kind of reality indeterminate but determinable in different singularities which would become necessary in the long run.
Peirce was proud of having found a way to represent all of that in his iconic logic, and so to put to work the common sense perception of identity in the modal logic of the Gamma graphs.
Now, how can this representation of identity be also the representation of "recognizing an identity"? Here comes the most intriguing part. Peirce could unify "identity" and "recognizing an identity" by virtue of the practical "scribing" on the sheet of assertion or on the multidimensional continuum. "Recognizing an identity," in EGs, means to draw a line of identity between two points, knowing that there is always a different, loose possibility. It is a "re-cognition" because our drawing of the line is not only a cognition of a single point, but it is an acceptance of the original identity of the two points: they are distant but the same. The drawing itself is our recognizing; in EGs there is no recognition without the actual drawing of the line, exactly as there is no triangle without its image. It can be a mental or a virtual act, but it has to be a kind of diagram within a general interpretation. It is what we call a "mathematical gesture."
It was enough for Peirce because his project was about representing analytically our synthetic reasoning, including the one we usually perform when we recognize objects that change all the time.
However, since our project differs from his in so far as we want to synthetically understand our synthetic reasoning, we have to find what shares the very characteristics of the line of teridentity in our everyday reasoning. Now, we learned from EGs what the properties we are looking for are: evidence, generalization, perfect continuity (reflexive, transitive, possible, and general).
What does the line of identity mean in everyday life and reasoning? What does it refer to in our common-sense synthetic logic? There are several passages in which Peirce compares the line of identity and the continuum to different psychological tools, but we are not concerned here with this sort of metaphorical comparison. (66)
We are not looking for a mere comparison or metaphor but for the actual realization of that synthetic reasoning in our everyday experience, a kind of reasoning that should show in flesh and bones our synthetic paths like the pragmatic maxim and the hypothetical detour (just to mention two important tools that Peirce discovered but did not express in a synthetic pattern). According to our paradigm, synthetic reasoning is a reasoning that recognizes identity within a change. So the question is: how do we reason synthetically? In other words, how do we recognize identities of the type A=B? More existentially, how do we put together the different steps of our logical reasoning?
Now, we have seen that--almost unknowingly--Peirce himself used a synthetic way to represent his analytic of reasoning. "Mathematical gestures" and "graphs" are the ways he used, and he gave them an enormous explicative power. Their rationale is to bring universals in particulars. Besides, we have seen what it means as far as logical identity is concerned: to recognize an identity means to draw a continuous line in a multidimensional plastic continuum.
What are the synthetic ways of reasoning in which we usually use universals in particulars? What are the ways of recognizing identities that display the characters of a "perfect sign" and "perfect continuity"?
The answer has to do with the synthetic tools Peirce himself used. Exactly like "mathematical gestures" and "graphs," our synthetic reasoning is a "gesture." Mathematical gestures and graphs are parts of a broader synthetic tool, which is "gesture" in general. What is a gesture? Gesture is any performed act which carries a meaning (from gero = to bear, to carry).
Generally speaking, we can say we really clarify something when we transform our vague familiar comprehension into a habit of action, and not when we have a good definition. Even more generally, when we say that a person's understanding of something needs to be tested, we mean that the effectiveness of a reasoning will be shown only in its synthetic feature.
However, we are not seeking just any gesture. All gestures carry meaning but they do not necessarily serve to recognize an identity. So, any gesture is meaningful, but not every gesture shows a synthetic reasoning. In order to understand "syntheticity" we have to look for a "perfect" or a "complete gesture," a gesture that respects all the characters we found following EGs.
We should call a "complete gesture" a gesture which has all the semiotic elements blended together almost equally. Now for a gesture to be complete, it has to be a general law (it has to have a general meaning) that generates replicas; it is actual when it indicates its particular object; and it expresses different possibilities of Forms and Feelings. (67) The three semiotic characteristics really describe what a complete gesture is and should be: creative because of possible forms and feelings, singular and unrepeatable in its individuality, recognizable for its unity and conformity to an established pattern.
Mathematical gestures and graphs are already an example, and everyday life furnishes countless instances. However, before seeing the everyday life examples, let us remember that abduction and pragmatic maxim already share many characteristics of our complete gestures. They work within, and thanks to, a continuity. They are "moving" forms of thought that follow in both directions (retro- or forward-) our continuous changing reality. In both reasonings we use particular gestures as method. In abduction we reconstruct the mental performance of a gesture, namely a kind of gesture involving the totality of the clues that a surprising phenomenon reveals. To get to hypotheses, whether existential or scientific, we have to imagine diagrammatically the gesture that brings together all hints we detect.
By definition, the pragmatic maxim already constitutes this very embodiment of meaning in mental or physical experiences--so much so that sometimes it has been mistaken for a sort of experimentalism. In any case, this error shows how much, according to the maxim, the totality of meaning requires a particular embodiment.
Complete gestures realize full meaning. Our dally experience is full of gestures that have already achieved this degree of clearness. Religious gestures are a good example of them because they are intended to be really meaningful: bathing for purification or baptism, genuflecting or bending or prostrating for prayers, pilgrimages to different parts of the world. As anthropology explains, ritual gestures fill up everyday life in education, love, and death. A gesture is not an action among others. It is the expression of a meaning embodied in one person at a singular moment, and tending to become a habit for the person and eventually for the generalized person, the people or the tradition. (68)
What are we performing in every complete gesture? On the one hand, a particular act performed by a particular person is embodying a general rule according to a certain interpretation. On the other hand, the person is here creating a "necessary" habit. The repetition of the Form involves a replica of the feelings, and it tends to foster a habit of action. Possibilities become acts and want to become habits or necessities. As a matter of fact, the same gesture can be accomplished by many different people, but it becomes actual only when a singular person is actualizing it, and only for that person does it tend to become a general habit.
In "gestures" we have that perfect continuity involving both generality and singularities that Peirce was unsuccessfully striving for on an analytic ground. Does the gesture respect the criterion of perfect continuity, namely generality, modality (plasticity), transitivity and reflexivity? We have already seen that gestures carry a possible and vague meaning that progressively gets determinate till a habit is established. So we can easily find in this movement generality, modality, and transitivity. Moreover, the possibility of learning a new gesture--like a new language (which is the performance of a gesture, as anybody verifies when learning a foreign language)--or a gesture replacing the one we habitually perform, corresponds to the open possibilities of the loose end in EGs. The semiotic constitution of gestures shows their reflexivity exactly as it does in the line of identity of EGs. Any part (law, actual realization, aspect) has the same properties of the general continuum affording alternative possible realizations and applications of the same gesture (that is, we can perform the same purification bath in many ways, modeling slightly different versions of the same meaning).
We can know that all properties form a "complete" unity because if any part is missing, gestures become merely formal. As we soon discover, meanings without gestures are dead, and gestures without meaning are empty. In other words, "complete gestures" are composed of possible meanings and possible realizations through a specific form that requires a singular individual. Every one of the three elements is purely possible, and it becomes more and more determinate the more the gesture gets accomplished. When it is not realized in each part of it, meaning comes back to bare possibility, while the gesture itself remains useless. It is what we call a formal gesture, and formal gestures can be mental gestures like reasoning, creative gestures like literary works, artistic gestures like painting or sculptures or musical creation, but there are formal gestures in religion, in love, and in death.
Now, performing complete gestures fashions identity. Identity is the continuity of those complete gestures, and we know that those gestures require neither formal repetition nor endless performance. Continuity exceeds that: perfect continuity is the continuity of meaning through gestures. In this sense performing a complete gesture means identifying an object.
Recognizing an identity through changes means being able to perform the same gesture. That is our synthetic way of understanding reasoning: being ready to act in the same way. It is also what is implied in our informal way of saying that we really understand someone when we can put ourselves in his shoes. This is the "real continuity" in identity. That very kind of empathy allows us to recognize identity through changes, and it is in difficult cases well expressed in figures such as Homer's Ulysses (recognized only through the scar), the resurrected Jesus of the Gospels (almost never recognized except through his gestures), or future selves in Dante's Divine Comedy. (69)
This is the synthetic way of proving pragmatism. Meaning is the sum of all our conceivable effects. We can understand that analytically like Peirce did by breaking down any singular piece of our reasoning from a mathematical, phenomenological, or semiotic point of view or we can understand it synthetically through complete gestures.
The argument we have provided follows Peirce's insight about the uncountable nature of continuity. If there is a continuity exceeding any multitude stemming from set theory, there must be also a reasoning exceeding the analytic pattern of the set theory. Since that exceeding continuity is the original reality we refer to in our reasoning about "sequence"--which is the kind of reasoning that both pragmatism and common sense follow--we should find a different path of reasoning, complementary to the analytic one and in communication with it.
Our new paradigm implies that synthetic judgment (and reasoning) is the judgment (and reasoning) that recognizes--with evidence--identity through changes. How can we perform this reasoning? We do it through complete gestures. Complete gestures--like the line of identity showed in EGs--save identity through changes because of their continuous semiotic nature.
We obtain analytical and vague reasoning by differentiation. Analytical reasoning loses the identity through changes because it is concerned with breaking up the identity, as its etymology suggests (analysis from analyo, discomposing, breaking up). As we said, it is very useful as well--and we made use of it also to understand what the characters necessary to a perfect gesture are--but it has another aim, that fosters precision, definition, and calculation, and that is very profitable in many cases but perfectly hopeless in many of our everyday businesses, and sometimes hopeless also--like in the case of discoveries or assessments of meaning--in many of our scientific activities.
A vague reasoning does not recognize an identity, but it is probably the richer state of our knowledge, the one in which our primeval beliefs lie. Of course, these beliefs are only vague, and thus are only secure within the limit of the need they point out but, sometimes, to ascertain our need is the first step toward any scientific decomposition and meaningful gesturing. (70)
University of Molise, Italy
(1) Joseph Brent, C. S. Peirce. A Life (Indianapolis and Bloomington: Indiana University Press, 1998). Note that Fisch's slips at the Peirce Project and the James-Peirce correspondence are a necessary complement to Brent's biography.
(2) On the difficulty of really finding the "proof of pragmatism," see the classic paper by Max Fisch. Max H. Fisch, Peirce, Semeiotic and Pragmatism: Essays by Max H. Fisch, ed. K. Ketner and C. Kloesel (Bloomington: Indiana University Press, 1986), 362-75. See also Christopher Hookway, "The Pragmatist Maxim and the Proof of Pragmatism," Cognitio 9 (2005): 25-42.
(3) Charles Sanders Peirce, The Essential Peirce, vol. 2, ed. The Peirce Edition Project (Bloomington-Indianapolis: Indiana University Press, 1998), 335; hereafter EP2.
(4) Charles Sanders Peirce, "The Bed-Rock Beneath Pragmaticism", Papers of Charles Sanders Peirce, MS 300, page 57. Manuscripts (MS) and letters (L) are collected in the Houghton Library of Harvard University and classified according to the catalogue R. S. Robin, Annotated Catalogue of the Papers of Charles S. Peirce (Amherst: The University of Massachusetts Press 1967). Hereafter, citations to this manuscript collection will be by manuscript (MS) number or folder of letters number (L) and page number.
(5) Charles Sanders Peirce to William James, 1 December 1902, Correspondence of Charles Sanders Peirce, L 224.
(6) See Thomas L. Short, Peirce's Theory of Signs (Cambridge: Cambridge University Press, 2007), 42-59; Max H. Fisch, Peirce, Semeiotic and Pragmatism, 184-200; Rosa M. Mayorga, From Realism to "Realicism": The Metaphysics of Charles Sanders Peirce (Lankam: Lexington Books, 2007).
(7) C. S. Peirce, Collected Papers, vols. 1-6, ed. P. Weiss and C. Hartshorne, (Cambridge Mass.: Harvard University Press 1931-1935) and vols. 7-8, ed. A. W. Burks, (Cambridge Mass.: Harvard University Press 1958). Hereafter, CP followed by volume and paragraph number. The quotation is taken from volume 1, paragraph 128. CP 1.128.
(8) Short, Peirce's Theory of Signs, 178.
(9) Andre De Tienne, "Is Phaneroscopy as a pre-semiotic science possible?" Semiotiche 2/04, 15-29.
(10) Charles Sanders Peirce to William James, 1 December 1902. Correspondence of Charles Sanders Peirce, L 224.
(11) Charles Sanders Peirce, Writings of Charles Sanders Peirce, vol. 2, ed. The Peirce Edition Project (Bloomington-Indianapolis: Indiana University Press, 1984), 267-68; hereafter W2. Hereafter quotations from the Writings will use the initial W followed by the number of the volume.
(12) See Robert Hanna, Kant and the Foundations of Analytic Philosophy (Oxford: Clarendon Press 2001), 191.
(13) See Jakko Hintikka, "C.S. Peirce's 'First Real Discovery' and Its Contemporary Relevance," Monist 63 (1980): 304-15.
(14) These latter had really the idea of a different pattern that would have been completely synthetic, including in this synthesis also the actual method and tools of reasoning. But they lacked Peirce's profound mathematical and semiotic understanding.
(15) "On a New List of Categories," in CP 1.545-559; "Questions Concerning Certain Faculties Claimed for Man," in CP 5.213-263; "Some Consequences of Four Incapacities," in CP 5.264-317; "Grounds of Validity of the Laws of Logic: Further Consequences of Four Incapacities," in CP 5.318-357.
(16) Charles Sanders Peirce, The New Elements of Mathematics (NEM), vol. 3, ed. C. Eisele (The Hague: Mouton, 1976), 774-5; hereafter NEM3. For these studies on continuity see both Matthew E. Moore, "On Peirce's Discovery of Cantor's Theorem", Cognitio 8, no. 2 (2007): 223-48 and Giovanni Maddalena, Metafisica per assurdo. Peirce e i problemi dell'epistemologia contemporanea (Soveria Mannelli: Rubettino, 2009): 137-92.
(17) Peirce, NEM3, 776.
(18) The chronology of changes is now pretty well established. For the latest solutions see Jerome Havenel, "Peirce's Clarification on Continuity," Transactions of the Charles S. Peirce Society 44, no. 1 (2008): 86-133 and Maddalena, Metafisica per assurdo, 193-224. The genesis of Peirce's continuum is well explained by Matthew E. Moore, "The genesis of the Peircean Continuum," Transactions of the Charles S. Peirce Society 43, no. 3 (2007): 425-69 and its possible mathematical development by Fernando Zalamea, El continuo Peirceano (Bogota: Universidad Nacional de Colombia, 2001).
(19) Peirce, CP 6.203-4.
(20) Peirce, CP 4.642; Charles Sanders Peirce, Manuscripts (MS) 204, 10-18.
(21) Peirce, CP 4.642.
(22) Zalamea subsumes transitivity under modality (Zalamea, El continuo Peirceano, 51-75), but I prefer to make it a general character of Peirce's continuum because of the fundamental role of transitivity in explaining "change."
(23) Kant, Critique of Pure Reason (Indianapolis, Cambridge: Hackett Publishing Company, 1996) A711, B739. Peirce states the same difference between mathematics and logic--and also stressed the primeval role of mathematics for logic--in the third Lowell Lecture (MS 458, 4).
(24) Peirce, EP2, 255.
(25) Peirce, EP2, 350-353.
(26) A good example of vagueness is the spontaneous belief in God, while a good example of the support that abductive inference receives from vagueness is shown by the idea of God in "A Neglected Argument for the Reality of God." Peirce, EP2, 434-50.
(27) For this distinction, see Paul Ricoeur, The Course of Recognition, trans. D. Pellauer (Cambridge: Harvard University Press, 2005).
(28) Sun-Joo Shin, The Iconic Logic of Peirce's Graphs (Cambridge Mass.: The MIT Press, 2002), 13-35. As is well known, icons, indices, and symbols represent the relationship with the dynamical object respectively by similarity, brute connection, and interpretation.
(29) Zalamea shows that EGs' iconic level better guarantees the representation of the mathematical reality on which logic relies, allowing for the representation with the same tools of more complicated and open logical patterns. Fernando Zalamea, Los graficos existenciales peirceanos (Bogota: Universidad Nacional de Colombia, 2010).
(30) MS 300, 52.
(31) MS 300, 23.
(32) Peirce, CP 4.530.
(33) Peirce, EP2, 22-224.
(34) Peirce, NEM4, 317.
(35) Peirce, NEM4, 313-30.
(36) Peirce, NEM4, 318--19.
(37) Charles Sanders Peirce, Pragmatism as a Principle and a Method of Right Thinking. The 1903 Harvard Lectures on Pragmatism, ed. P.A. Turrisi (Albany: State University of New York Press, 1997), 131.
(38) Peirce, EP2, 270n.
(39) For a complete explanation, see Short, Peirce's Theory of Signs, 264-70.
(40) Peirce, NEM4, 328
(41) Peirce, NEM4, 325.
(42) "It is agreed that a certain sheet, or blackboard, shall, under the name of The Sheet of Assertion, be considered as representing the universe of discourse, and as asserting whatever is taken for granted between the graphist and the interpreter to be true of that universe." Peirce, NEM4, 328.
(43) Peirce, CP 4.512.
(44) Peirce, CP 4.512-514.
(45) Peirce, NEM4, 330.
(46) Peirce, CP 4.406.
(47) See Roberts, The existential graphs of Charles S. Peirce, 51.
(48) Peirce, NEM 4, 330.
(49) MS 300, 48.
(50) Peirce, CP 4.445, figure 79.
(51) Peirce, CP 4.449, figure 83.
(52) Peirce, CP 4.561, figure 198.
(53) Peirce, CP 4.561.
(54) "[In the Gamma graphs] the line of identity must be totally abolished or rather must be understood quite differently. We must hereafter understand it to be potentially the graph of teridentity by which means there always will virtually be at least one loose end in every graph." Peirce, CP 4.583. Technically speaking, the loose end arises from the rule of Erasure and Iteration that "permits a branch with a loose end to be added to or retracted from any line of identity." Peirce, CP 4.505.
(55) "A Line of Identity that abuts upon a Cut, whereas on its Area or on its Place may look alike at its two ends; but an essential part of every diagram is the Conventions by which it is interpreted; and the principle that Graphs are Endoporeutic in interpretation, as they naturally will be in the process of scribing, confers a definite sens, as the French say, a definite way of facing, a definite front and back, to the Line." Peirce, NEM 4, 329.
(56) MS 300, 46-7.
(57) Peirce defines "aspect" as "the word which I propose to use as the technical designation of a qualisign that is so related to a sinsign." "So related" means that a qualisign may be a "fluctuating inconsistent memory of a sinsign, namely of a definite individual existent which is significant because of the circumstances of its existence." MS 284, 65-6.
(58) Peirce, CP 4.448.
(59) Peirce, CP 4.448.
(60) Peirce, CP 4.544.
(61) For the problem of the form in a general reconstruction of Peirce's semiotic, see James Jacob Liska, A general introduction to the semiotic of Charles Sanders Peirce (Indianapolis and Bloomington: Indiana University Press, 1996), 20-4.
(62) MS 637, 30.
(63) Peirce, CP 4.544.
(64) Peirce, CP 4.447.
(65) Peirce, NEM4, 317-18.
(66) MS 300, 42.
(67) Mead's idea that an index extrapolates a character from a gesture is here fully respected. See also George Herbert Mead, Mind, Self and Society (Chicago and London: University of Chicago Press, 1972), 95. My view agrees with Mead on many points, above all in considering language as a particular kind of gesture. However, this view gives to Mead's insight a logical foundation in continuity and a semiotic definition of gestures.
(68) I hope in this way to render justice to very powerful insights of authors like MacIntyre, Taylor and Colapietro. See Alasdair MacIntyre, After Virtue: A Study in Moral Theory (Notre Dame: University of Notre Dame Press, 1981); Charles Taylor, Sources of the Self (Cambridge: Cambridge University Press, 1988); Vincent Colapietro, "Toward a Pragmatic Conception of Practical Identity," Transactions of the Charles S. Peirce Society 42, no. 2 (2006): 173-205.
(69) Referring to Dante, Auerbach develops an idea of "figura" which really fits the kind of identity I want to propose. "Figural identity" will be the development of the gesture theory. See Erich Auerbach, Dante: Poet of the Secular World, trans. Ralph Manheim (New York: NYRB Classics, 2007).
(70) For this article I am very grateful to the Fulbright Program for the Research Scholar Grant 2009-10, to the Institute for American Thought, and to the Peirce Project that hosted me. In particular, I want to thank Andre De Tienne, director of the Project for his stimulating friendship, which is always part of my papers, and Anthony Graybosch whose suggestions, both philosophical and linguistic, have been almost as powerful and alive as his friendship.
Correspondence to: Giovanni Maddalena, Department of Human, Social, and Historical Sciences, University of Molise, via De Sanctis, 86100, Campobasso, Italy.
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|Publication:||The Review of Metaphysics|
|Date:||Mar 1, 2012|
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