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I am reading a paper on Brouwer's intuitionism. It mentions that according to Brouwer, the concept of continuum is perceived as a whole by intuition. However, it also mentions setting up choice sequences. If "continuum" is indeed perceived intuitively, why would we even attempt to use such choice sequence? We should automatically understand the notion "continuum". I fail to understand the relationship between choice sequence and intuition.

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  • TLDR Philosophy to many metaphilosophical theories consists of bringing intuitions to the formal light of language and logic, and then exploring the ramifications. Thus, defining infinity in terms of algorithms rather than a transcendental object, besides furthering metaphysics, also shows relationships that exist between mathematics, computer science, and logic, and goes towards seeing formal systems as a distinct topic as one does when adducing such claims as the Curry-Howard correspondence.
    – J D
    Commented May 9, 2023 at 21:04
  • "the concept of continuum is perceived as a whole by intuition" Correct: we "see" the geometrical line and that is the intuitive continuum, but mathematician tried to "build" it numerically: Cauchy, Weierstrass, Cantor. This historical process culminated with Cantorian set theory, that was unacceptable for Brouwer. Commented May 10, 2023 at 13:13
  • Thus, Brouwer attempted at a new "numerical" construction of the continuum (the natural numbers are acceptable for Brouwer): choice sequences. Commented May 10, 2023 at 13:15

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As you have read, mathematical constructivism rejects mathematical objects as entities, and allows for existential quantification in the case they can be constructed, or in a more modern parlance, computed. Thus, the debate on potential and actual infinity, to the constructivist, is no debate at all, since actual infinity is a Platonic entity that our intuition somehow supplies us with, and lacks actual rigor in terms of using a formal systems, which can be seen as an algorithm that computably (or recursively in older language) creates theorems using a rule set that itself includes rules of logic. So, a choice sequence can be understood as an algorithmic grounding of the term infinity.

Why do all of this? Well, first of all, infinity is a very useful concept, especially in defining continuity, and continuity is a purported property of space-time. But for a constructivist, it creates a problem in metaphysical explanation (SEP), because a constructivist rejects notions like Platonic realms and transcendental truth in favor for, in terms of my own discipline, physical computation (SEP). A Platonist can wave Plato's wand of woo and just summon forth 'infinity' and claim there is some other physical reality. Of course, that's an empirical no-no because it violates a basic precept that the burden of proof falls on the claimant, and without an adequate proof consisting of empirical evidence, one is simply left to wonder why the woo isn't just voodoo.

Thus, a constructivist, particularly a robust defender of the project of materialism that stems from a physicalist conception of the universe needs an explanation of what this useful infinity is. The alternative to all of the non-scientific alternate universes, modal realisms, and transcendental realities boils down to accepting some form of nominalism (SEP) and conceding that the term 'infinity' has a semantic grounding that doesn't exist with an empirical character, but rather a linguistic one. In fact, one can simply ground it in physical computation, and now there is a metaphysical explanation of infinity that is nominalistic, AND to boot a series of techniques for proving theorems and finding axioms.

For instance, in reverse mathematics, one of the chief concerns is computing enumerable sets from computable total functions (which is a type of infinite mapping composed of an infinite domain). These sorts of metaphysical strategies for semantic grounding of 'infinity' have wide applicability, not just in mathematics, but in computer science as well, where programing languages as virtual machines are often defined in terms of operational semantics.

So, while strictly speaking one can just use intuition, and rely on Platonic mathematical philosophy to metaphysically justify mathematics, in the same way Plato did 2,500 years ago, the curious mind rejects such mathematical philosophy as a fiction and certainly lacking in the rigor that Hilbert is chiefly famous for insisting guide mathematics (and one should add mathematical formalisms of computer science and mathematical logic). There is nothing wrong with pretending infinity exists and the natural numbers "just are" to get through calculus in high school, but at higher levels of math PA, TM, RCA0, ZF, and the full burden of formalized systems of mathematics look for a more complete explanation, and for many of us, one compatible with philosophy of science.

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