Abstract
According to L.E.J. Brouwer, there is room for non-definable real numbers within the intuitionistic ontology of mental constructions. That room is allegedly provided by freely proceeding choice sequences, i.e., sequences created by repeated free choices of elements by a creating subject in a potentially infinite process. Through an analysis of the constitution of choice sequences, this paper argues against Brouwer’s claim.
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Notes
This is the mainstream position today. If the modern concept of real numbers had been developed in his time, Plato would presumably also belong here.
See page 8 and chapter 2 of Brouwer (1907).
“In this way” is very vague. While some of the details will be fleshed out below, the phrase also reflects a vagueness and lack of detail in Brouwer’s own papers. See Kuiper (2004) for an attempt at filling in some of the details Brouwer omitted.
For more on this, see Hansen (2016).
In the second and third cases it should really be “an actually/potentially infinite equivalence class of actually/potentially infinite, converging sequences of rational numbers”. In the interest of avoiding cumbersome formulations, I will pretend that a sequence is a real number, rather than an element of one.
This is, of course, a view originating with Aristotle (1930).
Van Atten has informed me that he only intended to say that if the third element of \(\alpha \) has been chosen to be 1, then it is known that a choice sequence \(\beta \), for which something different from 1 has been chosen as its third element, is not equal to \(\alpha \).
One might deny that such a decision to identify really has the force to secure actual identity. But then, we would launch into an even more extensive disagreement with Brouwer, so I will not argue against it here.
I am overstating the simplicity a bit. If you and I each produce a choice sequence and we have so far, by chance, picked the same elements in the same order, and we both intend to expand our respective sequences according to the same restrictions, if any, we have nevertheless produced different sequences. (The two sequences will be equal (so far), but not identical. Brouwer also makes this distinction, for example in his definition of “species” (1952, 142). Troelstra (1977) makes the distinction using the terminology “extensional identity” and “intensional identity”.) That is not captured by “\(\langle 4,9,\text {intention to expand}\rangle \)”; there are also concrete facts about who the creator of the sequence is, when it was started, etc., that belong in a complete analysis of the constitution of a choice sequence. However, this complication is irrelevant to the issue at hand, for there is still no need to invoke the existence of indeterminate objects.
This operator has previously been employed by Niekus (2010) to help clarify aspects of intuitionism.
Please note that F here only disambiguates between potential and actual infinities of identity facts, and not between \(\mathbb {N}\) being potentially and actually infinite.
The branch of formal intuitionism in which I think it should have been most obvious that this conflation happens is in Beth’s semantics (Beth 1964, 444ff.), which is precisely an attempt at capturing the semantics of sequences of choices.
See Brouwer (1930) for an overview of what the freely proceeding choice sequences are supposed to contribute.
This demand on functions is closely related to what is demanded of real numbers; see the last few paragraphs of Sect. 1.
Actually, it is not essential to the argument that the condition would be satisfied in a platonic universe; it just makes for a nice contrast for the purpose of explaining the situation in the intuitionistic universe.
Note that none of this is intended to suggest that classical mathematics is the (or a) correct mathematics. It just means that if classical mathematics is not, then the intuitive notion of size can probably not be captured by the notion of cardinality. While one may take that as a reason to prefer classical mathematics, one can also go in the opposite direction and adopt the anti-Cantorian position that there is no (non-trivial) notion of infinite size to capture. As a third option, one may consider the notion to be primitive.
Hence, the assumption behind my concession above is also false: the two freely-proceeding choice sequences \(\langle 0,1/2,\text {intention to expand}\rangle \) and \(\langle 1,1/2,\text {intention to expand}\rangle \) are “duplicates” in the sense that they represent (at the given point in time) the same point on the continuum (and there is no atemporal fact about what point they represent).
Van Atten (2007) has reached the opposite conclusion. I find his line of reasoning to be extremely obscure. If one reads pages 89–93 in isolation, it would seem that van Atten reaches essentially the same conclusion as I did in Sect. 3 (although couched in the more flowery language of the phenomenological tradition). But he nevertheless agrees with Brouwer, based on a vague and unsubstantiated claim that the “inexhaustibility and non-discreteness of the (intuitive) continuum” somehow fits together with the undecidability of the extensional identity of freely proceeding choice sequences of intervals (p. 87). Apparently, the idea is that the use of nested intervals delivers the non-discreteness, but the classical real numbers can also be defined in terms of sequences of nested intervals, and they are discrete (i.e., the continuum is identified with a set of points). The difference to classical mathematics is supposed to be that the sequence is unfinished. But if it is unfinished, it has a last term (at any given time) and that last term represents an interval of positive length with rational endpoints. Further, it seems like the undecidability is supposed to match the inexhaustibility. That is, the (alleged) poverty of facts about whether pairs of sequences are identical or different is expected to deliver richness of ontological structure. I do not understand how.
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Acknowledgements
This paper originates in a dissertation chapter I first drafted more than eight years ago. I received valuable feedback on that draft from participants in the 4th French PhilMath Workshop, in particular Mark van Atten; in the 8th Scandinavian Logic Symposium; and in the Northern Institute of Philosophy’s 2012 Reading Party, as well as from Crispin Wright. More recently, I have also benefited from discussion with Claire Benn, Georgie Statham, Olla Solomyak, and Daniel Telech.
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Hansen, C.S. Choice Sequences and the Continuum. Erkenn 87, 517–534 (2022). https://doi.org/10.1007/s10670-019-00205-3
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DOI: https://doi.org/10.1007/s10670-019-00205-3