Abstract
It is known that the notion of the absolute ego of consciousness was described by Husserl in terms of the constitution and the continuous flow of internal time. It is remarkable that gradually Husserl tended to think of it not only in terms of temporality but in its generality as the source of all temporality, “reached” through a radical phenomenological reduction. We take into account the stages by which he was led to the impredicativity of the absolute ego of consciousness and try to demonstrate how this is reflected in the axiomatization of continuum in certain non-Cantorian mathematical theories to the extent that undertake a formalisation beyond natural intuition. We also review those theories’ approach to classical mathematical notions as this approach is more close to Husserl’s shift of the horizon idea.
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Notes
- 1.
Husserl did not clarify to the end the meaning of the absolute ego in general of his Cartesian Meditations (Husserl 1931) and has drawn criticism on the part of philosophers like Theodor Adorno who claimed that Husserl did not succeed of getting rid of a grounded Cartesian ego (Adorno 1982, Ch 4, pp 227–228).
- 2.
The term impredicativity is used here in the sense of impredicative mathematical theories in which there is no stratification of the mathematical universe and, intuitively speaking, one cannot comprehend (or describe) the elements or the parts but in terms of the whole, or a big part of it. It should be noted in addition that in platonic sense impredicativity of an object is the impossibility to assign to it any predicates at all, defined thereby identically as the non-being (Plato’s Parmenides, 163D, The Loeb Class.Library).
- 3.
A first-order formal language ℒ is one that, roughly speaking, allows quantification only over countably many elements of this language and does not allow quantification over higher order objects, e.g. sets or functions.
- 4.
“The totality of the group of original sensations is bound to this law: It transforms into a constant continuum (in ein stetiges Kontinuum) of modes of conscience, of modes of being-in the flow and in the same constance, an incessantly new group of original sensations taking originally its point of depart, to pass constantly (stetig) in its turn in the being-in the flow. What is a group in the sense of a group of original sensations remains as such in the modality of the being-in the flow.” (Husserl 1996, §38, p 102, transl. by the author).
- 5.
This problematic arises from the difficulty to describe ontologically under the same terms the continuum as a whole and its constituent unities. In Plato’s Parmenides the instantaneous change in the state of a physical body is attributed to the effect of a somehow intermediate state between rest and motion, the ′εξαíφνης, not expressible in spatiotemporal terms (Parmenides, 156D, E, The Loeb Class. Library), whereas in Aristotle’s On Coming to Be and Passing Away material points or lines are defined as limits, óρια, of material bodies which in their turn cannot be composed by points or attachments but by indivisible bodies or magnitudes (Coming to Be and Passing Away, 320b 15 and 316b 15, The Loeb Class. Library).
In R. Descartes, physical space in its infinitely divisible parts (up to extentional points) is defined as a primary substance, Res extensa, filled up with matter, as spatial extension is a substantial characteristic of matter (Discours de la Méthode, pp 168–169, Garnier Flammarion, Paris).
These extensional individualities are defined in Leibnizian monadology as incarnations of unique monadic localities representing in particular the body which they “affect” and whose entelechy realize. Space is thus, what results from those uniquely defined monads taken together (G. Leibniz, Fifth Letter to Clark, §47).
- 6.
“The Universe of living that composes the “real” content of the transcendental ego is not co-possible but as the universal form of the flux, a unity in which all particular elements are integrated as flowing by themselves. …We can see in them (the forms of the states of living) the formal laws of the universal genesis, according to which, thanks to a certain noetico-noematical structure, are constituted and united continuously the modes of flux: past, present, future.” (Husserl 1931, Fourth Meditation, pp 63–64, transl. by the author).
- 7.
In this approach we follow the lessons of 1893–1917. We leave aside, in this article, Husserl’s subsequent views on the matter putting in doubt the distinction between immanent and absolute time, see Die Bernauer Manuskripte (1997/1998), Hu XXXIII.
- 8.
The original German text under the title Formale und Transzendentale Logik. Versuch einer Kritik der logischen Vernunft was published in 1929 in the Jahrbuch für Philosophie und phänomenologische Forschung edited by E. Husserl, Vol. X, pp 1–298.
- 9.
Impredicative notions in mathematics are generally those in which the definiens uses the definiendum e.g. an open set (open interval) of the real line is not defined as the union of singletons (one-element sets) but in terms of other basic open sets (open intervals).
- 10.
In the sense that this evident generic similarity justifies the transposition of the eidetic relationships “discovered” in the universe of common intuition to that beyond this “horizon”. Although P. Vopěnka implicitly assumes this phenomenological principle in his Prolongation Axiom he seems to deny it in a later expository article on the philosophical foundations of Alternative Set theory where he allows for the possibility of a complete collapse of our intuitions beyond a genuinely qualitative shift of the horizon to apeiron (Vopĕnka 1991). But this eventuality contradicts with the Husserlian idea of our Life-world as gründenden Boden (grounding soil) of an ever shifting horizon.
- 11.
If F and G stand for functions, which in the AST extended universe are sets or classes, the Prolongation Axiom states that: For each countable function F, there is a set function f such thatF ⊆ f. It is important here to have in mind that countability of a function is, in fact, countability of a class of ordered pairs of elements and that a set function can be an uncountable set of ordered pairs of elements.
- 12.
In a formal context, AST works with sets and classes as objects. Sets are definite (might be very large) but sharply defined and finite from the classical point of view in the sense that in its universe of sets, AST accepts the axioms of Zermelo–Fraenkel system with the exception of the axiom of infinity. Classes represent indefinite clusters of objects such as the class N of natural numbers in the classical sense. In this context the notion of a semiset represents, roughly, blurriness and non-surveyability in the observation inside a very large set (see Vopĕnka 1979, Ch I).
- 13.
The indiscernibility equivalences \(\dot{=}\) are binary relations that are Π-classes (having the reflexive, symmetric and transitive property among others) and equipped in addition with the property of compactness; that is, for each infinite set U there is at least a pair (x, y) with x, y ∈ U such that, x≠y and \(x\dot{ =} y\). For further details and topological definitions of AST (monads, figures, closures, connected sets, etc) based on the indiscernibility equivalences (see Vopĕnka, 1979, Ch III, §1, 2, 3).
- 14.
In a less strict mathematical formulation the Axiom of Choice states that: Given a non-empty class of non-empty sets a set can be formed containing precisely one element taken from each set in the given class. Although the Axiom of Choice might strike someone as being intuitively obvious it may be less so if one has to deal with sets or classes of uncountably infinite cardinalities. An intuitive version of AC is produced by AST theorist A. Sochor concerning countable classes in AST sense (Lano 1993, p 152).
- 15.
Concerning the extensional development of nonstandard analysis, mainly A. Robinson’s version, one is led to the introduction of nonstandard elements endorsing, in effect, the Axiom of Choice or Zorn’s lemma in the use of free ultrafilters both in the construction of the nonstandard structures themselves and in the proof of significant theorems (e.g. Loś theorem, Mostowki collapsing function). Zorn’s lemma, in fact, is logically equivalent to the Axiom of Choice which in this context, in its stronger form of Global Choice, “induces” indirectly a notion of classical (actual) infinity. For more details, one is referred to Robinson (1966), Stroyan (1976) and especially to Connes et al. (2000) for a more intuitive presentation of the (uncountable) Axiom of Choice.
- 16.
For instance, regarding any object that can be described uniquely within internal mathematics as standard such as the set of real numbers ℜ, a real element x is infinitesimal in case for all standard ε > 0 we have | x | ≤ ε. Next, x≅y (x is infinitely close to y) in case x–y is infinitesimal and further, if E ⊆ ℜ and E standard, E is compact in case for all x in E there is a standard x0 in E with x≅x0. Regarding the definition of classical mathematical continuity, if f and x are standard then f is continuous at x in case y≅x implies f(y) ≅f(x), see (Nelson 1986, Ch 1, pp 2, 13).
- 17.
For a thorough development of the ideas of general topology based on the predicate and axioms of standardness in IST, we refer to (Diener and Diener 1995, Ch 6, p 109).
- 18.
Let λ-intervals be intervals of the form (\(\frac{\alpha } {{2}^{\nu -1}}\), \(\frac{\alpha +1} {{2}^{\nu -1}}\)). Then L.E.J Brouwer (1992, p 69) defined real numbers as follows:
We … consider an indefinitely proceedable sequence of nested λ -intervals \({\lambda }_{{\nu }_{1}}\), \({\lambda }_{{\nu }_{2}}\), \({\lambda }_{{\nu }_{3}},\ldots \) which have the property that every \({\lambda }_{{\nu }_{i+1}}\) lies strictly inside its predecessor \({\lambda }_{{\nu }_{i}}\) (i = 0, 1, 2, …). Then, by the definition of λ-intervals the length of each interval \({\lambda }_{{\nu }_{i+1}}\) at most equals half the length of \({\lambda }_{{\nu }_{i}},\) and therefore the lengths of the intervals converge to 0. (…) We call such an indefinitely proceedable sequence of nested λ-intervals a point or a real number.
It should be noted that the point P is thought to be the sequence \({\lambda }_{{\nu }_{1}},{\lambda }_{{\nu }_{2}},{\lambda }_{{\nu }_{3}},\ldots \) itself and not something as the limiting point (the unique accumulation point of the midpoints of these intervals) to which according to the classical conception these nested λ-intervals converge. Each of these \({\lambda }_{{\nu }_{i}}\) is considered then as part of the point P (Van Atten et al. 2002, p 212).
- 19.
By virtue of the definition of a spread as a “law on the basis of which, if again and again an arbitrary complex of digits (a natural number) of the sequence ζ (the natural number sequence) is chosen, each of these choices either generates a definite symbol, or nothing, or brings about the inhibition of the process together with the definitive annihilation of its result; … Every sequence of symbols generated from the spread in this manner (which therefore is generally not representable in finished form) is called an element of the spread. We also speak of the common mode of formation of the elements of a spread M as, for short, the spread M”, L. Brouwer formulated the Continuity Principle for the universal spread C as follows:
A law that assigns to each element g of C an element h of A (the natural numbers), must have determined the element h completely after a certain initial segment α of the sequence of numbers of g has become known. But then to every element of C that has α as an initial segment, the same element h of A will be assigned (see Van Atten et al. 2002, pp 222–224).
The principle of Open Data in its simplest form can be stated symbolically as follows: Aα → ∃n (α ∈ n and ∀ β ∈ n, Aβ) where A is a syntactical variable for any mathematical formula and α, β stand for lawless sequences. This principle essentially identifies under predication a lawless sequence α with all those lawless sequences of its neighborhood starting with the same initial segment n, see Troelstra (1977, §2.6, p 14). In a stronger than this, continuity principle, if one denotes by Cont LS the class of lawlike operations on lawless sequences assigning natural numbers to lawless sequences such that:
for Γ ∈ Cont LS, ∀ α ∃ x ∀ β ∈ < α0, \({\alpha }_{1}, \ldots, {\alpha }_{(\mathrm{x}-1)} > (\Gamma \alpha = \Gamma \beta ),\)
then:
∀ α ∃ x A(α, x) → ∃ Γ ∈ ContLS ∀ α A(α, Γα) see (Troelstra 1977, §2.6, pp 14–19).
- 20.
In Husserl’s phenomenology there is an analogy between mathematical and perceptual intuition in the sense that questions concerning mathematical intuition should be simply more specific cases of questions concerning intentional reference and directedness to objects. A counterargument is offered by those who insist on the difference between mathematical and perceptual intuition on the grounds that while in perception objects of intuition are determinate and individually identifiable, that seems to be what is missing in the case of mathematical objects e.g. in the mathematical intuition of the symbol ‘A’ standing for the empty set, see Tieszen (1984, pp 399–400).
References
Adorno T (1982) Against epistemology: A metacritique (trans: Willis Domingo). Blackwell, Oxford
Bernet R (1983) La présence du passé dans l’analyse husserlienne de la conscience du temps. Revue de métaphysique et de morale 2:178–198
Brouwer LEJ (1992) In: van Dalen D (ed) Intuitionismus. Bibliographisches Institut, Wissenschaftsvelag, Mannheim
Connes A, Lichnerowicz A, Schützenberger MP (2000) Triangle of thoughts (trans: Gage J). Editions Oedile Jacob, Paris
Diaconescu R (1975) Axiom of Choice and complementation. Proc AMS 51:175–178
Diener Fr, Diener M (1995) Nonstandard analysis in practice. Springer, Berlin
Drossos CA (1990) Foundations of fuzzy sets: a nonstandard approach. Fuzzy Sets Syst North-Holland 37:287–307
Husserl E (1931) Méditations Cartésiennes (trans: Emm. Levinas G, Peiffer). Librairie Armand Colin, Paris
Husserl E (1970a) In: Lothar Eley (ed) Philosophie der Arithmetik. Husserliana 12. M. Nijhof, The Hague, Netherlands
Husserl E (1970b) The crisis of European sciences and transcendental phenomenology (trans: Carr D). Northwestern University Press, Evanston
Husserl E (1970c) Logical Investigations, 2 vols (tran: Findlay JN). Humanities, New York
Husserl E (1973) In: Claesges U (ed) Ding und Raum: Vorlesungen. Husserliana 16. M. Nijhoff, The Hague
Husserl E (1982) Ideas pertaining to a pure phenomenology and to a phenomenological philosophy (trans: Kerten F). Martinus Nijhoff publishers, London
Husserl E (1984) Logique formelle et logique transcendantale (trans: Bachelard S) Editions PUF, Paris
Husserl E (1996) Leçons pour une Phénoménologie de la conscience intime du temps (trans: Dussort H). Editions PUF, Paris
Lano K (1993) The intuitionistic alternative set theory. Ann Pure Appl Logic 59:141–156
Lavine Sh (1994) Understanding the infinite. Harvard University Press, Cambridge MA
Longo G (1999) The mathematical continuum: from intuition to logic, ch 14, Naturalizing phenomenology. Stanford University Press, Stanford, CA, pp 401–425
Moran D (2000) Introduction to phenomenology. Routledge, New York
Nelson E (1977) Internal set theory: a new approach to nonstandard analysis. Bull Am Math Soc 83(6): 1165–1198
Nelson E (1986) Predicative Arithmetic. Mathematical notes. Princeton University Press, Princeton
Patočka J (1992) Introduction à la Phénoménologie de Husserl (trans: Abrams Er) Ed. Millon, Grenoble
Robinson A (1966) Non-standard analysis. North-Holland, Amsterdam
Stroyan KD, Luxembourg WAJ (1976) Introduction to the theory of infinitesimals. Academic Press, New York
Tieszen R (1984) Mathematical intuition and Husserl’s phenomenology. Noûs 18(3):395–421
Tieszen R (1998) Gödel’s path from the incompleteness theorems (1931) to Phenomenology (1961). Bull Symb Logic 4(2):181–203
Troelstra A (1977) Choice sequences. A chapter of intuitionistic mathematics. Clarendon Press, Oxford
Van Atten M, Van Dalen D, Tieszen R (2002) Brouwer and Weyl: the phenomenology and mathematics of the intuitive continuum. Philosophia Mathematica 10(3):203–226
Vopĕnka P (1979) Mathematics in the alternative set theory. Teubner-Texte zur Mathematik, Teubner Verlag, Leipzig
Vopĕnka P (1991) The philosophical foundations of alternative set theory. Int J Gen Syst 20:115–126
Weyl H (1977) Das Kontinuum (Italian Edition, care of B. Veit) Bibliopolis, Napoli
Woodin HW (2001) The continuum hypothesis I, the continuum hypothesis II. N Am Math Soc resp. 48(6):567–576 and 48(7):681–690
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Livadas, S. (2010). Impredicativity of Continuum in Phenomenology and in Non-Cantorian Theories. In: Carsetti, A. (eds) Causality, Meaningful Complexity and Embodied Cognition. Theory and Decision Library A:, vol 46. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3529-5_10
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