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If the abstract realm(s) are either completely independent of the physical or are ultimate causes, neither of these seem to push the abstract realm toward being so "contained" (say, having certain things and not others). At the very least I'm pretty sure we don't think of abstract realms as infinite randomness containing every pattern (as one example of a less restricted realm). This may fill the role of ultimate causes too; we are but one pattern. And being independent, their nature would not influence our nature, so again why “restrict” the abstract as we seem to do?

Do we do it to help mathematicians focus their thinking? That seems kind of selfish to limit the abstract for one/few purposes, but is that a reason why we seem to do so?

There are other non-abstract realms which are madness-like/infinitely random, but those are usually religious, not what I think of as ‘abstract’. Why is the abstract realm now (was it always) so “orderly” (has certain things and not others)?

Is this the state of affairs as I see it, what historically shaped it so? What right do we have to cast out certain abstract entities?

*(One thing which may add context and help answers is that I am skeptical of a non-empirical, solely mathematical reasoning capacity. If we do have non-empirical ideas and reasoning, it seems much broader than math, and not just fictional. I feel just as secure about other “truths” than just mathematical ones using my “abstract” thinking capacity.)

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    Unless you have in mind only math and exact sciences, I do not see how our "abstract realms" are anywhere close to orderly. They are a hot mess full of intractable complexity, nonsense and incoherence. Even when somewhat tamed by philosophers, at least continental ones. Derrida and Deleuze especially come to mind. And even math has "wild problems" that defy classification.
    – Conifold
    Jan 30, 2023 at 21:43
  • @Conifold I guess my worry is any limits we do place seem non-abstactly generated, like from concrete motivations or keeping math productive. I know there is a concrete limit of our finite nature, but that seems more of a limit of degree that can be navigated around. Only non-abstract human activities (not our finiteness) can permanently zone-out parts of the abstract realm from abstract thought. Finiteness may preclude all-at-once abstract thinking, but does not rule out any specific abstract object or family of abstract objects.
    – J Kusin
    Jan 31, 2023 at 1:18
  • I still do not follow what sorts of limits you think we place that concern you. Pragmatic? That practically useful anything is heavily favored hardly calls for special explanation, and it is not for historical reasons or out of courtesy to anybody. The same goes for not pursuing what is unproductive. We are not just finite, we also evolved to survive and multiply. I am struggling to think of anything without an obvious trivial explanation. What are those non-mathematical abstract truths that you feel secure about, but which are left out for non-obvious reasons?
    – Conifold
    Jan 31, 2023 at 1:50
  • @Conifold Limits imposed through philosophy itself or Socratic methods. Like from plato.stanford.edu/entries/fictional-entities/#OrthUnorNeoMein weren’t Zalta and Wolterstorff arguing for huge adoptions of abstract objects like Meingongian and fictional objects? Are they in or out? Even if they’re not restricted and in, does that force other abstract things out? For a traditionally left out abstract object, I feel quite secure using a fictional-turn-abstract “wise lion Leo” say to learn about knowledge and wisdom, which are at least partly abstract.
    – J Kusin
    Jan 31, 2023 at 7:12
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    The boundary is vague, but I think we can reasonably distinguish philosophers introducing new abstracta (as Kant did with noumena, Meinong with non-existents, etc.) and the sort of meta bookkeeping in analytic philosophy where the already familiar abstracta are recast as abstract objects, fictions, tropes or whatnot. By the way, Maddy argued that mathematicians follow MAXIMIZE maxim in their constructions (which is why self-restrictive conceptions like intuitionism and predicativism are consistently rejected by most) that clashes with ontological parsimony favored by philosophers.
    – Conifold
    Feb 1, 2023 at 0:30

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Note in advance: I am assuming that the abstract realm you're referring to is something along the lines of the "third realm" that Frege posited (alongside physical and mental domains).

Philosophers vs. early set theorists: these are not really very distinct categories. To some extent, we can distinguish between a trend in analytic philosophy and a trend in early, mainline set theory, though, one with expansionist aspects and one with restrictionist ones. This will be my core point of departure in what I write from here.

From Alexus Meinong to David Lewis: so first note that Alexus Meinong, in the apparent camp of the analytic philosophers, did have a rather unrestricted abstract realm to his viewpoint's name. As they say:

Not only existing things but also all kinds of non-beings (nonentities, Routley 1980: 7) find their place in Meinong’s all-embracing theory of objects — among them even impossible objects, like the round square, as well as paradoxical, “defective” objects, like some forms of the liar [the liar sentence] or special cases of purely self-referential thinking (the thought about itself, for example).

Meinongianism has reemerged and evolved into modern talk of "impossible worlds", which are supposed to be yet possible and actual (as worlds, though their inhabitants are not possible). But the most jarring purported impossibilia, i.e. true contradictions, are not admitted so much among plenitudinous Platonists at least so far as it is said:

If every consistent mathematical theory is true of some universe of mathematical objects [emphasis added], then mathematical knowledge will, in some sense, be easy to obtain: provided that our mathematical theories are consistent, they are guaranteed to be true of some universe of mathematical objects.

One version of this kind of Platonism preserves Meinong's round square, without meaning to actualize a contradiction, by having the round square as an object that encodes incompatible properties without exemplifying them: the round square is not actually round or square, much less both at once.

A rather different doctrine in the same extended family (the family of "plenitudinous" theories, in this case a plenitude of concreta), David Lewis' modal realism, was cited by Brian Greene (the string-theory popularizer) as a philosophical background for the string-theoretic landscape of zillions of "universes," but whereas Lewis estimated the number of modal-realist universes to be among the transfinite cardinals, Greene's string theory "merely" involves approximately 10500 universes. But so even Lewis' multiverse seems fairly restricted (the specific cardinal he guessed at is usually calculated to be radically infinitesimal when compared to the bounty of higher set theory). Now, the Lewisian multiverse is, again, a case of concreta, except insofar as one might interlock Edward Zalta's encoding-vs.-exemplifying doctrine of abstracta with Lewis' indexical theory of actuality (so that we have an indexical theory of abstracta). But even aside from considerations of such a fused theory, we might style the relatively nonactual Lewisian worlds as functionally equivalent to abstracta from within any given such world. At any rate, generic possible worlds (conceptually disjoint over the bare Kripkean sense of these things or hypostasized in a Lewisian manner) and their ensemble aren't seemingly so restricted; and then impossible worlds are even less so.

Why were many early set theorists concerned to prove the well-ordering principle? Recall that Cantor did not believe that "all the (particular) sets" were contained in some general set, but he thought that the divine intellect (which he did understand in terms very much like Kant's talk of intellectual intuition) was what they were all elements of, and this intellect was explicitly stated to be a particular (the individual personality of the divine nature).

But, parallel to Frege and ever since, most set theorists who have spoken of a universal set or universal class or other such notions, have represented this entity rather impersonally (though note the perpetual talk of "witnesses" who "think" various things in set theory, to this day). More pointedly, there was, early on, a major push to establish the well-ordering principle. Why?

By itself, Cantor's original diagonal argument about the lack of a bijection between ℕ and ℝ only showed just that: a difference in "size," not a determinate relation of lesser-and-greater. Cantor separately came up with the series of the ℵa (which essentially does obey a uniform lesser-or-greater commandment) and then advanced the Continuum Hypothesis, but it was an open question whether |ℝ| would be one of the ℵa. Were it not, then transfinite arithmetic might've been even easier to represent than if (G)CH were true; the expression "20" would just go to some independent symbolism, "κ0" say, and then "2κ0" could've just gone to "κ1," and all the κb would be their own series, and we would have not detained ourselves with the intricacies of trying to overlap the aleph and kappa numbers.

But oftentimes the emotionally-minded reason for mathematicians to "invent" a new kind of number is to use it in calculations that connect it to other kinds of numbers. It was easy enough (eventually) to get the imaginary unit and use it to keep the symbolism for real numbers with the symbolism for complex numbers, and so to facilitate the arithmetic for going back and forth from ℝ-terms solely to ℂ-terms inclusively. It took a lot longer to do something similar with infinitesimals (I think we had to wait on Abraham Robinson's nonstandard analysis for this, and NA is still not as symbolically "friendly" as John Conway's surreal infinitesimals), but we still did it. So for transfinite arithmeticians, the desire was to keep the terms of their arithmetic connected up enough so that the successor operation on the zeroth aleph, which accesses ℵ1, would "be close friends with" the powerset operation which accesses |ℝ|. But then we had to "prove" the well-ordering principle, so that the series of powersets and the series of successors, among the alephs, would involve commensurable terms, and we could definitively write expressions like, "ℵ1 ≤ |ℝ|."

The introduction of the axiom of foundation had to do with Russell's paradox, albeit not absolutely directly. If all sets are not elements of themselves, and this due to an axiom as such, the very question of a set that would be and not be an element of itself is not quite so "askable" in the first place, and the paradox is dismissed on the level of preliminary semantics. However, the relationship between the foundation axiom and the Russell paradox is thematic more than expressive of a technical requirement, so though restricting the abstract realm of mainline set theory to well-founded sets was par for the course for years on end, Quine and then Aczel loosened the restriction on parafounded sets among mainline set theorists. And now, especially in the light of the Hamkins multiverse, we have broken those chains and melted many of them down.

Now, the foundation axiom is separate from the well-ordering principle. You can have sets that are well-ordered but circular (or which head up infinite descending elementhood sequences) as well as noncircular/nondescending but not well-ordered. Still, I suspect that whatever feeling we have for transfinite arithmetic (at least in the sense of the desire that there be a nifty system/method of this arithmetic) depends very much on both principles being in play. A series of parafounded alephs would be hard to commensurate (maybe impossible to commensurate) with the well-founded ones in the tidy way envisioned by Cantor. It would be like ending up with the κb all over again, maybe.

Yet so when Paul Cohen proved that ZFC cannot decide which aleph the Continuum's cardinality is, he was able to undermine the going-back-to-Cantor hope for a reliable/stable system of transfinite arithmetic. Two years later, Paul Benacerraf identified the identification problem; roughly around the same time, Robinson devised nonstandard analysis; and so attitudes and viewpoints like game-theoretic formalism and general pluralism were on the up and up again. The now-prevailing conception of set-theoretic multiverses is a manifestation of analytic-philosophical ensembles of possible and impossible (and transpossible) worlds in the sense that Hamkins, Gitman, and so on are the inheritors of plenitudinous Platonism in a way that is (sociologically speaking) very much like how Brian Greene inherited much of his motivation to believe in the string-theoretic multiverse from another plenitude-theoretic analytic philosopher (Lewis).

Conclusion: so, among the community of analytical philosophers (from Frege and Meinong onward), there was a less insistent demand that the abstract realm be subject to complicated restrictions on its content (noncontradiction was not even held to too fast "in the beginning" and is even less mandatory nowadays). For their own peculiar sentimental (aesthetic) reasons, set theorists concerned to develop a system of transfinite arithmetic like unto the arithmetic for finite numbers that we know and love were keen to introduce restrictions into their abstract realms, until the time of Cohen and Benacerraf came and that keening (metaphorically) turned to wailing and gnashing of teeth. Then the less-restricted abstract (and concrete) realms current in the theorizing of analytic philosophers dovetailed with the "universe of sets" more, until now we have that neither family of abstract-realm doctrines (philosophical or mathematical) are so distinguished, but Plato's heaven has married Cantor's paradise, and their children are the multiversal narratives (and metanarratives) popular in physics and fiction nowadays, too.

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    I appreciate the detail, seems like you identity a lot differing motivations as to why the abstract realm changes through history and thinker to thinker. I guess I strongly indetify with Meinong's wide arms at the moment. thanks!
    – J Kusin
    Jan 31, 2023 at 1:03

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