I've always thought that the mathematical community's rejection of intuitionism should be thought of as a choice -- not a bad choice but still one we could've made differently. It's not incoherent, and possibly not even as hard to work with as most of us seem to think. It just reflects different intuitions about what mathematical objects are and how they should be treated. So it's really interesting to see someone be explicit about this -- this guy is revitalizing part of Brouwer's ontology, which is related to a sort of idealized theory of mind, against the more popular Heyting position that intuitionism should be thought of as a variant logic.

Intuitionist totality is a fundamental concept in intuitionist mathematics that sharply contrasts with the classical approach. While classical mathematics embraces the law of the excluded middle—the principle that every statement is either true or false—intuitionism demands a more constructive framework. A set is deemed totally defined within intuitionism only when a concrete method exists to determine membership for any given element. This necessitates a specific procedure to prove whether an element belongs to a set or not. Intuitionism is profoundly skeptical of potential infinity, requiring that all set elements be individually specified or constructed. This constructive approach has profound implications for fields like analysis, logic, and set theory. As L.E.J. Brouwer, a pioneer of intuitionism, asserted, "The real numbers are not to be thought of as given in advance, but rather as continually becoming." This quote encapsulates the intuitionist view of mathematics as a dynamic process of construction. In contrast, David Hilbert, while critical of intuitionism, acknowledged its significance, stating, "Intuitionism is a serious attempt to establish mathematics on a new foundation."

"...Lacan's invocations of intuitionism were intended to highlight the constructive nature of the symbolic order of meaning and discourse only. They were not meant as an endorsement of the underlying ontology. .... ..the intuitionist preclusion of actual infinity is not relevant to the meaning of Lacan's concept of the pas-tout" Russel Grigg: Lacan, Language and Philosophy 2008 - as discussed by Amy Swiffen Int J Semiot Law 2012 25 289-292 "From an intuitionist perspective to claim something with certain properties exists is to claim that an object with those properties can be constructed. Put differently, the truth of a statement is taken as equivalent to the mathematician being able to intuit the statement. ... Thus, for intuitionism, infinity is only potential"