Abstract
In this article I try to articulate a defensible argumentation against the idea of an ontology of infinity. My position is phenomenologically motivated and in this virtue strongly influenced by the Husserlian reduction of the ontological being to a process of subjective constitution within the immanence of consciousness. However taking into account the historical charge and the depth of the question of infinity over the centuries I also include a brief review of the platonic and aristotelian views and also those of Locke and Hume on the concept to the extent that they are relevant to my own discussion of infinity both in a purely philosophical and epistemological context. Concerning the latter context, I argue against Kanamori’s position, in The Infinite as Method in Set Theory and Mathematics, that the mathematical infinite can be accounted for solely in terms of epistemological articulation, that is, in the way it is approached, assimilated, and applied in the course of the construction of mathematical hierarchies. Instead I point to a subjectively constituted immanent ‘infinity’ in virtue of the a priori as well as factual characteristics of subjective constitution, underlying and conditioning any talk of infinity in an epistemological sense. From this viewpoint I also address some other positions on the question of a possible ontology of the mathematical infinite. My whole approach to the question of the infinite in an epistemological sense hinges on the assumption that the mathematical infinite subsumes the infinite of physical theories to the extent that physics and science in general deal with the infinite in terms of the corresponding mathematical language and the specific techniques involved.
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Notes
The term immanent, widely used in phenomenological texts, can be roughly explained as referring to what is or has become correlative (or ‘co-substantial’) to the being of the flux of one’s consciousness in contrast to what is ‘external’ or transcendent to it. For instance, a tree is transcendent as such to the consciousness of an ‘observer’ while its appearance in the modes of its appearing within his consciousness is immanent to it.
The phenomenological notion of life-world can be described in rough terms as an indefinitely extensible horizon of our special reduction-performing co-presence in the world, this latter meant as the primitive soil of our experience. A major work in which this notion is further elaborated is Husserl’s well-known Crisis of European Sciences and Transcendental Phenomenology (Husserl 1976b).
By eidetic laws or eidetic attributes in the world of phenomena one can roughly communicate to a non-phenomenologist what relates to the existence of objects or states-of-affairs as regularities by essential necessity and not by mere facticity. One may also consult E. Husserl’s Ideas I, (Husserl 1976a, Engl. transl., pp. 12–15).
“Die Unendlichkeit der Welt ist diese Offenheit, diese motivierte Möglichkeit neuer Erfahrungen und erfahrender Ausweisungen. Diese Unendlichkeit, hier als äussere gedacht, hat ihr Gegenstück in der inneren Unendlichkeit, die zu jedem einzelnen Realen gehört. Sie ist begrenzt durch den Umfang meines ‘Ich kann’. Der Raum ist als realer konstituiert mit meinem jeweiligen Allseitig-ihn-erschliessen-können, und dieses Können hat seinen eigenen subjectiven Horizont, der zwar nicht voll exakt umgrenzt ist, aber doch einen endlichen Limes hat.” (Husserl 2001, p. 164). Transl. of the author: “The infinity of the world is this openness, this motivated possibility of new experiences and determinations through experience. This infinity, here thought of as external, has its counterpart in the inner infinity which belongs to each particular real being. It is bounded by the scope of my ‘ I can’. The space is constituted as a reality through my respective all-around possibility of its unfolding and this possibility has its own subjective horizon which is not bounded in an exact sense, yet it has a finite limit.”
See, e.g., Livadas (2019).
An explicit reference is found in the first paragraph of Sect. 3.
A noematic object, a phenomenological term, is an object as meant constituted by certain modes as a well-defined object immanent to the temporal flux of a subject’s consciousness. It can then be said to be given apodictically in experience inasmuch as: (1) it can be recognized by a perceiver directly as a manifested essence in any perceptual judgement (2) it can be predicated as existing according to the descriptive norms of a language and (3) it can be verified as such (as a reidentifying object) in multiple acts more or less at will. More in Husserl’s Ideas I: (Husserl 1976a, pp. 229–232).
Temporal succession is thought of in the Philosophy of Arithmetic as the psychological pre-condition for the built up of number meanings and generally of the meaning of multiplicities (Husserl 1970, p. 28).
This obviously refers to relations between formal-ontological objects, including such mathematical objects as sets, classes of sets, functions, domains of functions in the form of Euclidean or non-Euclidean spaces, etc.
Hellman and Shapiro have developed, in Hellman and Shapiro (2012, 2013), a formal axiomatical system to construct a ‘regions based’ one-dimensional continuum which follows the Aristotelian credo that continua are not composed of points (as spatially postulated). However they do not follow Aristotle in a very important aspect, namely in that they make essential use of the actual infinite rejecting the notion of potential infinity espoused, at least on epistemic grounds, by Aristotle and many other mathematicians and philosophers since then.
The possibility of reaching an infinity in thought in the sense of immanence which does not correspond to a spatiotemporally founded one is also implied by the statement that “not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought” (Greek orig. of the underlined: διὰ γὰρ τὸ ϵ̓ν τῇ νοήσϵι μὴ ὑπολϵίπϵιν) (Ross 1960, Engl. transl., p. 43).
In An Essay Concerning Human Understanding, Locke argued about the ‘negativity’ of the idea of infinity on the grounds of the finiteness of human mind for which an infinite extension (over time) cannot but produce a negative idea of infinity, that is, a confusing and indeterminate one out of insufficient understanding (see Jacquette 2001, p. 66).
Even though Locke denied that we might have any positive idea of infinite number, space, or time he nonetheless claimed that the ideas of ‘causal power’ and ‘unity’ are found to be ubiquitous within experience (Garrett 1997, p. 18). As argued in the present article, unity (but not causality) can be integrated in the phenomenological sense of immanent infinity.
In what Brouwer described as the primordial intuition of mathematics intuitive continuum is the “substratum, divested of all quality, of any perception of change, a unity of continuity and discreteness, a possibility of thinking together several entities, connected by a ‘between’, which is never exhausted by the insertion of new entities” (Brouwer 1907, p. 17 in: Van Atten et al. 2002, p. 205).
These specifically phenomenological terms can be roughly described as a priori forms of intentionality toward an original impression just passed-by to the past (retention) and an a-thematic ‘expectation’ to the future (protention). For more the reader may look at Husserl (1966).
For a review of the fundamentals of the intuitionistic version of mathematical foundations, in particular the ad hoc continuity principles, the reader may consult (Heyting 1966).
The interested reader may look at Livadas (2020) for a detailed presentation of the issue.
Forcing theory is, roughly speaking, an ingenious mathematical technique initiated by the American mathematician P. Cohen in 1963, and further developed to a full-fledged mathematical theory, mostly motivated by the interest to resolve certain key mathematical questions such as the Continuum Hypothesis. More can be found in Cohen’s (1966) own work, and Kunen’s (1982).
It is questionable, though, whether the translation of a mathematical theory, be it a pure or applied mathematics theory, to the axiomatic set theory is indeed an ontological reduction properly meant or just a reduction to the meanings and methods associated with the semantics and the syntactical structure of set-theoretical models. Of course a supplementary reduction on the level of primitive logical-mathematical ideas such as an indefinite collection of abstract objects, the non-logical predicate of belonging, \(\in\), etc., may lead to questions pertaining to the nature of primitive mathematical objects themselves, but then whether one chooses an ontological interpretation in a platonic context or a subjectively based one is a matter of choice and sound argumentation.
Of course mathematicians treat infinite in everyday mathematical practice as an ideally existing yet concrete and complete mathematical object (or rather such a collection of objects), part of the predicative universe of a formal theory. This was acknowledged as far back as Aristotle’s Physics (Ross 1960, Engl. transl., p. 50). Yet this conventional attitude, dear to logical positivism, in no way affects the question of a possible ontological treatment of the infinite.
An interesting, though no further discussed on a subjective level, view of the mathematical continuum is E. Belaga’s in Halfway Up To the Mathematical Infinity I: On the Ontological & Epistemic Sustainability of Georg Cantor’s Transfinite Design. Belaga suggests that any phenomenologically and ontologically faithful axiomatization of the continuum should include a non-locality postulate (in the sense this term is understood in quantum information processing) to formally account for the following property of the continuum: All ‘points’, or ‘elements’ of the continuum are non-locally, i.e., simultaneously and at any moment, accessible. This non-local accessibility extends to all well-defined ‘slices’ (subsets in a set-theoretical terminology) of the continuum (Belaga 2009, pp. 34–35).
For structures \(\mathcal {M}_{0}=<M_{0},..>\) and \(\mathcal {M}_{1}=<M_{1},..>\) of a language \(\mathcal {L}\), an injective function \(j:M_{0}\longrightarrow M_{1}\) is an elementary embedding of \(\mathcal {M}_{0}\) into \(\mathcal {M}_{1}\) (\(j:\mathcal {M}_{0}\prec \mathcal {M}_{1}\)) iff for any formula \(\varphi (u_{1},....,u_{n})\) of \(\mathcal {L}\) and \(x_{1},..,x_{n}\in M_{0}\)
$$\begin{aligned} \mathcal {M}_{0}\models \varphi [x_{1},....,x_{n}]\;\;\text{ iff }\;\; \mathcal {M}_{1}\models \varphi [j(x_{1}),....,j(x_{n})]. \end{aligned}$$A class M is an inner model iff M is a transitive model of ZF under the \(\in\) predicate and contains the class of all ordinals, i.e., \(ON\subseteq M\).
In fact the elementary embeddings (into inner models) approach has its inherent limitations by virtue of Kunen’s fundamental result in Kunen (1971). Kunen has proved, capitalizing on the application of AC and the Erdös–Hajnal theorem, that if j is an elementary embedding from V into V then j must be the identity mapping. See Kunen (1971, pp. 407–408).
A subset A of the set of reals R is universally Baire if for all topological spaces \(\Omega\) and all continuous functions \(\pi : \Omega \rightarrow R\), the preimage of A by \(\pi\) has the property of Baire (a purely topological property) in the space \(\Omega\).
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Livadas, S. Is There an Ontology of Infinity?. Found Sci 25, 519–540 (2020). https://doi.org/10.1007/s10699-020-09669-x
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DOI: https://doi.org/10.1007/s10699-020-09669-x