Abstract
This paper deals with Husserl’s idea of pure logic as it is coined in the Logical Investigations (1900/1901). First, it exposes the formation of pure logic around a conception of completeness (Sect. 2); then, it presents intentionality as the keystone of such a structuring (Sect. 3); and finally, it provides a systematic reconstruction of pure logic from the semiotic standpoint of intentionality (Sect. 4). In this way, it establishes Husserlian pure logic as a phenomenological epistemology of mathematical logic.
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Notes
Labelled in this line as ‘constitutive structuralist’ (Hartimo 2012, pp. 102–103) the task of phenomenology is then to found epistemologically mathematical logic by uncovering the acts of mind (know as ‘consciousness’) in which the mind-independence of mathematical entities is constituted in mathematical symbolic thinking and knowledge.
See also the Prolegomena, paragraphs 65 and 66 entitled ‘The question as to the ideal conditions of the possibility of science or of theory in general. A. The question as it relates to actual knowledge’ and ‘B. The question as it relates to the content of knowledge’, respectively.
The role of these axioms is to secure the application of the operations (for instance, if the combination ‘\(+\)’ is used in a formal domain, then there must be pairs of elements a and b combinable in the form \(a+b\) and at least an x such that \(a+b = x\)).
In this case no supplementary determinations can be added to the characterization of the formal domain by its system of axioms, and it must be possible to identify univocally, via the axioms, each elements of the domain: such domains are ‘determinate’-defined domains.
Since it is anything but clear that Husserl’s notion of (defined) manifold can be understood as that of model in modern model-theoretic sense (i.e. as tuple consisting of a domain of objects and relations, possibly functions, on the domain), such expansions do not seem to be correlated with model extensions (obtained by adding individuals to a model domain and extending the relations on them); as I said above, my guess is that manifolds represent class of models (Husserl 2003, pp. 477–478, 482–483 [see also: pp. 488–489]) and consequently, that such extensions are correlated to extensions of models classes (gained by adding axioms to the theory).
In Formal and transcendental Logic, both are put together in the following terms: ‘How far does the possibility extend of “enlarging” a “multiplicity”, a well-defined deductive system, to make a new one that contains the old one as a “part”?’ (Husserl 1969, p. 97).
In short: ‘In the case of a mathematically definite manifold, the concepts of “true” and “formal-logical consequence of the axioms” are equivalent; and so are the concepts of “false” and “formal-logical anti-consequence of the axioms”’ (Husserl 1982, p. 164)—yet for the record, Husserl’s definiteness is still a matter of interpretative debate with no consensus for the moment (see the references in note 4, and also Centrone 2011; Gauthier 2004; Hill 2000; Isaac 2015 among others).
Namely, on one hand, the category of independence that characterizes autonomous entities (i.e. entities not necessarily integrated into wholes—these are “complete” entities), and on the other hand, the category of non-independence that characterizes non-autonomous entities (i.e. entities necessarily integrated into wholes—these are “incomplete” entities).
In this process, the simple inputs comprise autonomous and non-autonomous entities (see note 12) (both treated as parts in the process of their combination); and in case of compatibility, they produce a whole by complementation (i.e. a unitary complex of parts connected via a relation of foundation in the whole) (Husserl 1970b, LI III: Sect. 23, LI IV: Sect. 10).
These two compositional criteria successively determine the very possibility of any semanticity, and then condition the possibility of any semantic valuation (Husserl 1970a, LI I: Sect. 29) (Sect. 3.3)—on the side of meanings, this anticipates to some extent the distinction between formation and transformation rules (see Isaac 2016).
In other word, its triple stratification is an indirect answer to the problem of imaginary in mathematics (Sect. 2.1); still, the grounding relation remains here clearly dichotomous (dealing distinctively with the two sides of pure logic), and the possible syntactic-semantic correspondence of its dichotomized components is not at all yet justified (see Claim, page 6).
Viz. the one in which a physical sign is linked to some mental states or contents via psychological association (Husserl 1970a, LI I: Sect. 6).
The term of model in collocation with that of semiotic will signify here the schematization of the structure that links a sign to what it designates (viz. its reference via/or its signified when it works as a signifier).
So, unlike the classic semiotic triangle (sign, sense, reference) in which the meaning referential process articulates ‘two distinguishable sides’ (sign-sense and sense-reference) via a pivotal point (the sense) (Husserl 1970a, pp. 198–199) the sense does not constitute in Husserl’s model a step from the sign towards its signified object (i.e. an intermediary third realm of objects situated in some topos ouranos): it only consists in the process of linking immediately the expression to its object (directly given in the expression as meaning-act), and there is no medium obstructing meaning as vector of intentionality, provider of referential relation.
Indeed, they only occur in cases of realized references with the filled intentions of ‘complete expression[s]’ (Husserl 1970a, p. 192) (i.e. when the act of aiming at something is correlated with the act of hitting the targeted object, here intuitively given or imaginarily presented); and in this regard, they correspond to a narrower concept of intention, strictly correlative and distinguished from the wider one (somehow irrelative) that is involved in cases of unrealized references with the empty intentions of “‘mere” expression[s]’ (ibid.) (i.e. when the act of aiming does not point to any corresponding fulfillment by the targeted object, here only meant without intuitive givenness or imaginary presentation) (Husserl 1970b, LI V: Sect. 13).
Working here as unified idealities of act-species, they turn the referential orientations into stable objective source-to-target relations (determining the transcendent objects’ quids).
Here comes the influence of Lotze’s interpretation of Plato’s doctrine of Ideas.
In such unified idealities of act-species, the realized references are turned into logically basic relations of more or less adequate fulfillment (making the objects’ quomodos known).
In sum, it is nothing but the meaning-intentional analogon of a fulfilling intuition in the context of the intuitive mode of intentionality (Husserl 1970a, LI I: Sect. 14).
On this process, cf. Husserl (1970b, LI VI: Chap. 4 [especially: Sects. 40, 43, 48, 58]).
In fact, the resulting categorial domains are formal since they are made of materially undetermined variables of formalized contents, and analytic since they are governed by laws independent from the values taken by the variables in a formula.
In view of Definition 2, formalizing abstraction is expressly distinguished from the simple term of abstraction used to mean the extraction of a non-independent part of content.
Analogues of the existential axioms of the theoretical level of pure logic (see note 6), the laws of existence involved here nonetheless differ from the logical-mathematical case by their categoriality (i.e. instead of needing to be laid down on a case-by-case basis, they are valid for every value of the categorially typed variables of a meaning category).
Such a determination is processed through an intensionalization of the referential meaning: the syntactical efficiency of (formal linguistic) signs is assimilated into the signitive-significative intention (as vector of the apophantic categorial morphology), and the syntactic process of calculus fixes the sense of (formal) languages as intentionally-driven (Husserl 1970a, LI I: Sect. 20 [cf. Husserl 2003, pp. 491–492])—the prevalence of meaning over fulfilling-sense is thereby justified, while the fulfilling-sense itself becomes independent from any realized references.
Indeed, while it categorizes the intentional relation of some linguistic signs to their objects, the meaning-intention determines the categorial forms of objects (Sect. 3.2); and when such forms are then themselves taken as (formal) objects, they work as the intuitive counterparts of the formalizing abstraction by which they are produced, and provide as such the basis for the ontological formal-analytic domain.
In the Sixth Investigation: ‘If we are asked what it means to say that categorially structured meanings find fulfillment, confirm themselves in perception, we can but reply: it means only that they relate to the object itself in its categorial structure’ taken as the ‘categorial forms in the [...] objective sense’ (Husserl 1970b, pp. 280/307).
This process is precisely underpinned by the specific (categorial) case of signitive-significative intentionality in which the intuition of apophantics and the apophanticity of intuition are jointly performed in the process of fulfillment (Sects. 3.1, 3.2)—in fact, while it both determines meaning as Meinung and conditions the possibility of any givenness, the ontologization of the apophantic categorial morphology requires here fulfilling intuitions (as the matter of the intentional acts vectors of categories), and conversely, the fulfilling operativeness of any intuition necessitates (with respect to the categorial intentional acts) the categorization of intuitions proper.
Cf. Husserl’s conception of logic as the science of ‘the pure [and ideal] “laws of thought”, which express the a priori connection between the categorial[-essential] form of meanings and their objectivity or truth’ (Husserl 1970a, p. 225).
Yet, for a suggestive insight in that direction, see Woleński (1997, pp. 156–157) (and note 3).
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Dedicated to Claire Ortiz Hill, with Honor and in Friendship
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Isaac, M.G. Toward a phenomenological epistemology of mathematical logic. Synthese 195, 863–874 (2018). https://doi.org/10.1007/s11229-016-1249-z
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DOI: https://doi.org/10.1007/s11229-016-1249-z