Abstract
Phenomenology: where the phenomenological background of the entire work is presented and discussed.
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Notes
- 1.
Husserl believed that indexicals too had meanings, which he called “essentially occasional”, for they depend on the occasion and circumstances of enunciation (see 1st Logical Investigation, chap. 3).
- 2.
The term haecceity translates Duns Scotus’ haecceitas or “thisness”, which translates Aristotle’s τὸ τί ἐστι.
- 3.
Husserl 1962.
- 4.
A name refers rigidly if it refers to the same object independently of how it is conceived, i.e. if naming by that name goes along with an identifying intention. In this case, we usually say that the name is used consistently.
- 5.
Whereas for Frege identity is a relation among connotations, for Husserl, it is the object of a supervening act of keeping the same object (sometimes a mere something) under the intentional focus of acts with different intentional meanings. Whereas, for Frege, identities express the ego-independent fact that the same objects can be denoted by different connotations, for Husserl, identities are correlates of acts of identification, involving the ego and intentional consciousness in an essential manner. Identities are constituted, not simply “grasped” in identity-assertions.
- 6.
Intentional experiences are also called acts for the reason that the ego, who undergoes the experience, does not only provide the locus where the experience simply “happens”, but is actively involved in making it happen. I will also refer to the ego by an “it” since it is not always an individual person.
- 7.
“‘Real’ [is] that which exists in space and time” (Husserl 2006, p. 16). Abstract objects are ontologically dependent objects, such as the color of a body as a real aspect of it. Ideal objects are non-real objects.
- 8.
In fact, the real world simply is the maximally consistent system of all possible objectively valid perceptions.
- 9.
The physical object does not exist because its adumbrations are consistent; consistency is not a definition of existence. It is existence that implies consistency of adumbrations; therefore, one can take consistency as a reliable criterion (or sign) of existence (criterion = necessary condition). One can, however, advance the following definition of existence for physical objects: an object of the empirical world exists if, and only if, the ideal infinite system of all ideally possible perceptions of it is consistent. This definition can be generalized. Consequently, the existence of a real object is always sub judice, our practical and scientific lives must cope with this fact.
- 10.
“In contraposition to nature, to the world of factual spatial-temporal existence, to the ‘empirical’ world, there are, as one says, ideal worlds, worlds of ideas, which are non-spatial, non-temporal and unreal. And yet, they exist indeed […]” (Husserl 2006, p. 16).
- 11.
A terminological observation. Husserl uses the term “positing” acts to acts whose objects exist with the sense of being with which they are posited. Perception proper, for example, is in this sense a positing act, whereas misperception is not. They stand in opposition to non-positing acts, such as fantasying or daydreaming (when I entertain the phantasy of, say, an unicorn, conceived as a physical object, the unicorn, although existing in phantasy does not exist as the real object it is meant to be – phantasizing does not really posit its object; it does not confer real existence to it). Also, Husserl calls “objectifying”, in opposition to non-objectifying, acts whose objective correlates are objects in a restricted sense of the term, such as naming and judging (for Husserl, names denote individuals and judgments, states-of-affairs). I will not strictly adhere to Husserlian terminology. I use the term “posit” (also mean and intend) here as a generic term for intentional presentation, with its different degrees of “clarity”, i.e. intuitiveness, modes and characters.
- 12.
We must be very careful with the expression “in principle”. As I use it here, it does not mean “effectively” or “actually”. By saying that a problem is in principle solvable I only mean that it is not a priori, considering only meaning, seen as unsolvable.
- 13.
Mathematical objects, as I will argue below, are (ordinarily or “classically” posited as) abstract (ontologically dependent), ideal (non-real) objects outside space and time. To treat them otherwise, as, for instance, temporal objects, in the manner of intuitionists, is to falsify the experience in which they are posited. Which does not mean that experiences of constitution of the intuitionist type are illegitimate; on the contrary, all positing intentional experiences are legitimate on their own terms. My point is that intuitionism does not coincide with ordinary mathematics; it is a completely different thing. It cannot count as a philosophy of mathematics, only as an alternative conception of mathematics. My approach, in short, offers not only a possibility of philosophically clarifying usual, ordinary mathematics, but alternative versions of it too.
- 14.
There are many points in common to mathematics and phenomenology, both are eidetic, not factual sciences (sciences of essences, not facts) and both are a priori. In Ideas I (Husserl 1962 §§ 71, 72) Husserl raises a question that seems, then, natural: could phenomenology be mathematized? His conclusion is that it cannot. For one, phenomenology, as a material eidetic science, does not belong to the formal eidetic sciences like formal mathematics. Could, then, phenomenology be put together with material mathematical sciences, such as, for example, geometry? Can phenomenology be developed as a sort of geometry? Still, it cannot, according to Husserl, for geometry is an axiomatized and ideally logically complete (definite, in Husserl’s jargon) theory, which proceeds essentially by logical derivation from axioms, i.e. fundamental laws of essence, whereas phenomenology is a descriptive eidetic that is not and cannot be axiomatized. By being essentially non-formalizable and incapable of axiomatization, phenomenology is, for Husserl, essentially non-mathematical.
- 15.
Much has been debated about Husserl “transcendental turn”, which happened in between the publication of his Logical Investigations (1900–1901) and Ideas I (1913), more precisely in courses of the period 1906–1907, and what it means. Intentionality was central to his thought both before and after the turn. As I see it, however, before, in the “realist” period, Husserl’s goal was to investigate intentionality as a natural phenomenon within a naturalist context; after, in the “transcendental” period, he approaches intentionality as a pure form in a transcendentally purified context (the notion of epoché that I will examine soon is fundamental in this transition). Transcendental phenomenology imposes itself the task of investigating the necessary features of intentional experiences and intentional consciousness in general. The transcendental intentional subject is absolute, the center from where meaning flows; it is a function rather than a thing. In the transcendental period, intentionality is no longer seen as “a manner of seeing” things that may exist otherwise with a sense of their own. Nor, on an epistemological key, as the way in which the subject approaches, as knowing subject, the object of knowledge (the intentional object), which exists, with a sense it intrinsically has, independently of being known. Transcendentally considered, no object exists independently of being intentionally meant and no object has a meaning without being given a meaning in an intentional experience.
- 16.
See the detailed analyses in his Husserl 1970.
- 17.
The mereological sum a and b includes, as an abstract moment, the syncategorematic and, the categorial element of the objectual complex.
- 18.
Husserl attributes to Descartes the discovery of transcendental philosophy and the transcendental ego, the Cartesian ego cogito. But, according to Husserl, Locke took possession of the transcendental ego and psychologized it. However, he believed, only by being thoroughly de-psychologized the transcendental ego can serve scientific philosophy. Transcendental phenomenology is, for him, such a philosophy.
- 19.
The problem of intersubjectivity is a central problem in Husserlian phenomenology, to which Husserl dedicated the fifth of his Cartesian Meditations, part of Ideen II and Zür Phänomenologie der Intersubjektivität. The ego is primarily my ego; after the transcendental reduction, the primordial ego has the task of constituting the world and other intentional egos. Only after the rights of alter-egos are recognized, the ego can extrapolate the limits of individuality.
- 20.
The theme of the individual responsibility was dear to Husserl. In his last work, Crisis of European Science and Transcendental Phenomenology (Husserl 1954a) this topic is central. The “crisis” alluded to in the title is precisely one of responsibility.
- 21.
See Crisis.
- 22.
“Phenomenology is, as Husserl depicts it in his 1907 lectures, an eidetics of cognition. The method of reduction signifies the critical means of access not to any de facto consciousness but rather to the essential structural correlation of consciousness and objectivities per se intended therein” (Sandmeyer 2009, pp. 75–6).
- 23.
“Reduction” means, literally, to lead back (re-ductio). By using the term (cognate of the German verb “to reduce”, reduzieren) Husserl probably meant a going back to the intentional phenomenon as such.
- 24.
See Husserl 1960, §8
- 25.
The title of §8 of Husserl 1960 is precisely “The ego cogito as transcendental subjectivity”.
- 26.
In his article for the Encyclopedia Britannica (Husserl 1927), Husserl characterizes the “transcendental problem” as “having to do with the being-sense of ‘transcendent’ relative to consciousness”. The “transcendental attitude” is required so the phenomenologist can raise and deal with the “transcendental problem”. In the pre-epoché “natural attitude”, sense of being is a given, it does not require constitution; it does not have a genesis. At best, constitution has the epistemological sense of the unveiling of the object to the subject. The correlation object-for-the-ego/object-positing-ego avoids a plethora of ontological and epistemological problems originated in the naturalistic separation of subject and object.
- 27.
In the introduction to his translation of Husserl’s essay of 1936 “The Origin of Geometry” (Husserl 1954b) Jacques Derrida presents an interesting discussion of this question.
- 28.
However, whereas for Frege denotation requires connotation, for Husserl, as I interpret him, intentional directness does not depend necessarily on any particular attribution of intentional meaning.
- 29.
Check definitions in the third Logical Investigation (Husserl 2001).
- 30.
Husserl calls parts the independent components of a whole and moments its dependent components.
- 31.
What is the difference between an object, say a, and the singleton {a} whose sole element is a? Materially, of course, there is none, but formally there is a difference, namely, the categorial aspect of {a} that a does not have. Husserl calls “categorial” the abstract (ontologically dependent) aspects of objects of higher-order cognitive intentional acts, set-collecting in this case, denoted by {…}. Another important higher-order act is that which posits a state-of-affairs, for instance, “that the paper is white” based on the object “white paper”; “that the paper is white” is a content of judgement, not merely perception.
- 32.
It is worth noticing that not all abstract objects are ideal, although all ideal objects are abstract. Abstract objects are ontologically dependent objects, which all ideal objects are, since they are ontologically dependent on their realizations. For example, the number 2 would not exist if collections of two things did not exist. Abstract objects, on the other hand, can present themselves as aspects or moments of real objects, like the color or the form (the real, not the geometric idealized form) of physical bodies. Hence, abstract objects can be real, although they are never concrete, which are ontologically independent objects that can exist independently of the existence of other objects.
- 33.
Usually, this is how one defines cardinal numbers. In modern mathematics, where set theory provides a context of materialization (or instantiation) of ideal entities, the number 2 can be defined as the class of all pairs, or a particular set, {0, 1} or {{0}}, indifferently, provided these “avatars” have the same formal properties of the number 2 (they have, of course, other properties, but they are arithmetically irrelevant). Set theoretical avatars represent ideal objects only to the extent that they offer a particular material basis for abstraction and idealization and thus for intuiting the idealities they represent (abstraction can be understood in this case as the specification of which features of set theoretical representatives are and which are not arithmetically relevant).
- 34.
Empathy, that is, intending other egos (alter egos) as intentional agents, plays an important role in the process of objective positing. I cannot enter the theme here, but it is an important one when considering the constitution of an “objective world”, for instance, physical nature. I will come back to this when discussing the constitution of the modern notion of physical reality.
- 35.
Among the idealities that play “a universal role for a pure analytics” Husserl mentions “the fundamental form of the and-so-on, “the form of reiterational ‘infinity’”, which has “its subjective correlate in ‘one can always again’” (Husserl 1969 § 74).
- 36.
Husserl calls “nominal” the acts of naming and judging, whose objective correlates are, respectively, the thing named and the states-of-affairs asserted.
- 37.
Husserl developed the distinction between sense and denotation independently of Frege. In fact, both worked within a rich philosophical tradition in which this distinction was, in some form, already present, sometimes more, sometimes less clearly (see Husserl’s 1st Logical Investigation).
- 38.
Let us consider a more illustrative example, the positing of “the largest prime number”. The intentional meaning “largest prime number” is consistent with itself, for nothing in the definition of prime number rules out that there could be a largest one. However, the concept of prime number, once considered more comprehensively in the larger context of mathematics, requires that there is no such number. That is, the positing is externally inconsistent. The distinction between internal and external consistency seems to me necessary so conjectures (either true or false) have a place in mathematics. The existence of meaningful but false conjectures (such as, for example, “there is a largest prime”) requires the distinction between internal and external consistency.
- 39.
Since Kant did not accept intuitions-that, only intuitions-of, he did not explain convincingly how constructions, which are always particulars, can have general validity. How, for example, the construction that brings to light the fact that the internal angles of a particular triangle add to two right angles can justify asserting this property of all triangles?
- 40.
It is a task for formal logic, logical grammar in particular, to determine which terms are logical. The main feature of the logical being universality.
- 41.
Formal ontological categories are those that apply to objects merely as such, without further material specifications. See Logical Investigation VI, chap. 8.
- 42.
Bunge 1986.
- 43.
For Husserl, the infra-conscious levels of perception, closer to the sensorial given, are not, for not being fully conscious, strictly speaking intentional. But when higher-level intentional acts such as abstracting or judging are involved, perception is a truly intentional act.
- 44.
In her “Frege’s Attack on Husserl and Cantor” (Hill and Rosado Haddock 2000, pp. 95–107), Claire Hill argues that, in fact, through Husserl, Frege is in fact aiming at Cantor.
- 45.
The intuition of the ideal, however, does not require perception necessarily; we could have imagined a red flag instead of perceiving one. Imagination is also a form of presentification and can in the intuition of the ideal substitute perception proper.
- 46.
This has an important consequence; phenomenology can safeguard a Platonist way of seeing without embracing Platonism as a theory. Some phenomenologists have called this non-naïve Platonism. I call it, as I already did, Platonism between brackets, “Platonism”.
- 47.
One takes an object as a mathematical object when one attributes to this object only the properties of the mathematical object one takes it to be (even if it has these properties only approximately). But I am not particularly concerned with this question; I only want to emphasize the fact that whatever “taken as” is, it is an intentional act.
- 48.
As is clear, form-preserving (congruent) transformations constitute a proper subclass of triangularity-preservation transformations.
- 49.
Aristotelian empiricists might approach the matter from a different perspective. Suppose a denotes a triangular figure (a physical triangle) and P a property of a. By definition, P belongs to a as a mathematical triangle if for any physical object x, if x has P, then x is also triangular. Now, one can define a mathematical triangle as the “equivalence class” of all (roughly) congruent triangular figures. Any P that belongs to an element of the equivalence class as a triangle belongs also to all elements of the class as triangles, for they are all (roughly) congruent. The mathematical triangle that this class represents has all and only the properties that any element of the class has as a triangle. The empiricist can then take the mathematical triangle as only a façon de parler. But this would be a falsification of the mathematical experience. Mathematics posits ideal objects as ideal objects, not merely as ways of referring generically to physical objects. Neither Plato nor Aristotle are completely right or completely wrong. Geometrical forms are (with Plato) ideal, but (against Plato) they are not ego-independent. Geometrical forms are abstract aspects of actually or possibly existing physical objects (with Aristotle), but (against Aristotle) ideal forms have a sense of existence that is not that of real entities, although they may be or are, as in the geometrical case, constituted in acts whose matter are real objects.
- 50.
For detailed analyses of the relations between judging and experiencing (perceiving, in particular) see Husserl 1973.
- 51.
This is why it is so easy to change the question as to the meaning of an object into the question as to meaning of the word that denotes it, as analytic philosophers do. But whereas phenomenologists only need to inquire the phenomenon, analytic philosophers must step outside the intentional experience and inquire linguistic usage. See da Silva 2016b.
- 52.
See da Silva 2016b.
- 53.
See Husserl’s 1st Logical Investigation, entitled “Expression and Meaning”.
- 54.
We must allow also for signs that denote directly by convention and, I believe, indexicals, such as “this”, “that” or “I”, which have, for Husserl, a meaning that, however, is only completely determined in a context of use.
- 55.
See Husserl 1969, §74.
- 56.
- 57.
See da Silva 2005.
- 58.
Not as a matter of fact, but of principle. Self-consistency is a necessary criterion of existence – nothing exists that can support contradictory attributions; this is part of the meaning of existence, any type of existence.
- 59.
If the positing of either the object a or the object b is consistent, internally and externally, but that of both a and b is not, the ego is free to posit either a or b, but not both. The extension of the intentional meaning of the domain by the introduction of, say, a into it is valid if consistency is maintained, and only until it is maintained. Should an inconsistency follow from introducing a into the domain, then a does not belong there, but b may. If the domain of objects in question is posited as objectively complete, the fact as to whether a belongs or not to the domain is objectively decided (but maybe neither subjectively nor logically decided, in which case the decision stands as an ideal).
- 60.
See for instance Ideas I § 142 for the intimate connection between consistency and existence. The issue is related to an important question in transcendental logic, the justification of the principle of non-contradiction, related to the intentional positing of the concepts of being and reality.
- 61.
Benacerraf’s error was to presuppose that if numbers are sets they must be definite sets. For one, numbers are not sets, they can only be represented as sets. Moreover, numbers can be represented set-theoretically in any convenient way provided the representation is throughout consistent. In short, 2 can be (interpreted as) either {{0}} or {0, {0}}, although it is neither. Benacerraf purported to show that since there is no definite way of identifying numbers to sets, numbers are not sets, and must then be something else. I agree that numbers are not sets, but not for this reason. Numbers, as we will see later, are an altogether different type of objects, but they are objects, which can be individually intuited, referred to, named, and conceptually characterized.
- 62.
When discussing the idealizing presuppositions behind the principle of identity in his Formal and Transcendental Logic, Husserl recognizes the “I can always come back to this object in future experiences” as the noetic correspondent of the noematic meaning “object that can present itself again to me in future experiences”. The principle of identity is, for Husserl, as we will see later, rooted in the intentional positing of objects conceived as capable of manifesting themselves as the same in different intentional experiences.
- 63.
See Derrida 1989.
- 64.
Note that a non-apodictic truth is not necessarily an apodictic falsity.
- 65.
The intuition of the object is not per se a truth-experience; it is necessary that the presentation of the object, with the sense it has, fulfills or fails to fulfill, partially or completely, explicit or implicit expectations. Of course, the ego can, by reflecting on an intuition, judge with clarity, that is, truthfully; in such cases, intending and intuiting are concomitant, but intending is still there, as part of the reflexive act.
- 66.
A quote from Husserl seems in order here. Talking about the positing of a transcendent, in this case real world, he says: “What is transcendent is given through certain empirical connections. Given directly and with increasing completeness through perceptual continua harmoniously developed, and through certain methodic thought-forms grounded in experience, it reaches ever more fully and immediately theoretic determinations of increasing transparency and increasing progressiveness. Let us assume that consciousness with its experimental content and its flux is really so articulated in itself that the subject of consciousness in the free theoretical play of empirical activity and thought could carry all such connections to completion (it would be necessary to consider the mutual comprehension with other egos and other fluxes of experiences); let us further assume that the proper arrangement for conscious-functioning are in fact satisfied, and that as regards the course of consciousness itself there is nothing lacking which might in any way be required for the appearance of a unitary world and the rational theoretical knowledge of the same. We ask now, presupposing all this, is it still conceivable, is it not on the contrary absurd, that the corresponding transcendental world could not be?” (Ideas I, § 49). These considerations are, mutatis mutandis, valid for positing in general. To the extent that the positing is consistent and remains so in the continuous flux of experiences, the posited object exists.
- 67.
See FTL II Chap. 3.
- 68.
One must be very careful here not to shock those who are incapable of abandoning the natural for the phenomenological attitude and can easily misinterpret the whole thing in naturalist terms. Numbers do not exist unless they are consistently posited and rational beings can very well exist who do not posit them, but as long as, for us, numbers exist, they exist as they are meant to exist.
- 69.
In his “The Origins of Geometry” (Husserl 1954b) Husserl offers a model analysis of the constitution of geometrical objects and geometry from both the noetic and the noematic perspectives. The exposition of the series of noetic acts, with their noematic correlates, which may or may not be discernible in the factual history of geometry, constitutes what Husserl calls the “transcendental history” of geometry. Factual history records the traces (or a selection of them) that transcendental history leaves in culture.
- 70.
See for instance Nodelman and Zalta 2014.
- 71.
In the preface of his Das Kontinuum (1918), Weyl says that “it is not the purpose of his work to cover the ‘firm rock’ on which the house of analysis is founded with a fake wooden structure of formalism – a structure which can fool the reader and, ultimately, the author in believing that it is the true foundations” (Weyl 1994, p. 1). I too believe that formalism cannot account for the true philosophical foundations of anything.
- 72.
Husserl characterizes object in the sense of formal logic as “any possible subject of true predicative judgments” (Ideas I §3). Husserl’s characterization is equivalent to defining object as a subject of a meaningful assertion.
- 73.
Husserl gives as examples of objective categories those of Object, States-of-Affairs, Relation, Connection, among others (Husserl 2001, vol. 1, Prolegomena to Pure Logic §67).
- 74.
For Husserl, formal logic contains also, parallel to formal ontology, the discipline of formal apophantic logic, concerned, according to him, with syntactic categories to the same extent that formal ontology is concerned with ontological categories. For Husserl, the most fundamental task of apophantic logic is to investigate the a priori laws of meaningful combination of syntactic types. There is, for Husserl, a strict parallelism between syntactic and ontological types (the syntactic type Subject corresponding to the ontological type Object, Predicate to Property, and so on). Logical-grammatical laws determines the boundaries of formal meaningfulness for assertions, and correspond, on the ontological side, to the a priori laws regulating ontological types. Husserl’s logical-grammatical laws are Carnap’s “laws of formation” or the “rules of formation” of modern logic. Unlike modern logic, however, Husserl saw an ontological correlate to syntax that, however, is not still semantic in the modern sense. (In Formal and Transcendental Logic, Husserl introduces semantic notions, such as truth, among others, in apophantic logic; this clearly indicates a distinction between syntactic and semantic notions at the interior of apophantic logic itself, but nothing of the sort of Tarskian semantics.)
- 75.
Since G is a first-order theory, the formal truths valid for all groups (i.e. the formal properties of G) are theorems of G.
- 76.
As we know, these notions are logically independent.
- 77.
The exegesis of Husserl’s approach to this problem is difficult and occupied many researchers. See da Silva 2016a and the bibliography therein.
- 78.
This is what Husserl meant by saying that phenomenology is a descriptive material a priori science; phenomenology cannot do without direct intuition on its domain of investigation, the phenomenon of intentionality. But some material sciences, such as, for example, physical geometry are able to obtain a strong enough basis of intuitive truths to afford giving up intuition altogether.
- 79.
Essentially, this is what Husserl called the problem of “imaginaries” in mathematics. See da Silva 2010.
- 80.
“Knowledge of the objective-scientific world ‘is founded’ in the evidence of the life-world” (Crisis §34).
- 81.
As just said, this is a recurrent interest of Husserl’s; in Experience and Judgment, for example, he traces judgments back to perceptions, and in Crisis he carries out a detailed analysis of the intentional production of mathematized physical nature from perceptual empirical nature.
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da Silva, J.J. (2017). Phenomenology. In: Mathematics and Its Applications. Synthese Library, vol 385. Springer, Cham. https://doi.org/10.1007/978-3-319-63073-1_2
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