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Though the SEP article on abstract objects has long weighed on my mind, I never formed much of an opinion about the question until now. The opinion I did have was negative: the ~space/~time definition was unsatisfactory, because we then have the question of abstract space and abstract time to consider anyway (i.e. the Platonic Forms of space and time, if you will; or space and time as themselves "Forms," even).

But as I was thinking about generality-particularity as being subject to a transitive ordering, I wondered if we could posit that an object counted as abstract not because it was strictly general, but because its degree of generality (in the ordering) exceeded some kind of threshold.

Would this threshold be vague? Take an "obviously" mathematical statement, like 222 = 16. This is already somewhat "formal" or "general," but it also seems somewhat particular or material. So one step up in abstraction might involve generalizing over one of the terms, replacing it with a variable. Eventually, we can replace all the numbers (even the implied hyperoperator index for the exponentiation operator's dual appearance) with variables. We could continue to "abstract away" details of the resulting structure, perhaps indefinitely (say by shifting from hyperoperators to arbitrary operators). At what point is this mathematical statement going to fully turn into a logical one? Especially once we factor in things like Boolean algebra and category (or type) theory, demarcating mathematics from logic starts to resemble a sorites situation.

Now if mathematical objects are already abstract objects, this would be more like another vague threshold in a domain already separated from another one by a similar vague threshold. I suppose this might be related to higher-order vagueness, in which the vagueness of a vague threshold is itself an abstract boundary ("within" the "Form" of vagueness, so to say). Or perhaps the moral of the story is that logical objects (if you will) are fully abstract, physical objects are fully concrete, and then mathematical objects occupy the vague boundary between logic and physics in turn.

"Mathematizing" the threshold

The transitivity of the generality-particularity ordering seems in tension with circular such orderings. That is, if A was more general than B, B more than C, and then yet C more than A, well, I will at least say it is hard for me to understand what that would mean. But on the other hand, an infinitely ascending and descending such ordering seems possible, to me. Now, one way to represent the infinite descent (I think Aczel's hyperset theory rules this out, but I'm not sure about other hyperset theories) would be to take the surreal counterpart of ORD and then represent ORD - a, for all ordinals a and their surreal counterparts. This "terminates" at ORD - ORD = 0. So, in surreal terms, we can use every negative hyperoperation to describe an exotic surreal number using a transfinite ordinal base, e.g. the square root of omega, or more importantly for present purposes omega divided by 2. That indicates something like ORD/2, then. This "object" is also indicated by the mirror equation between an antifounded descending set and the ascending universe of well-founded sets, i.e. ORD - (ORD - a) = a, which leads to the idea of the point where ORD - A = A, which is just ORD/2.

Even more importantly, surreal infinitesimals can be formed at every transfinite ordinal scale, such that between any two surreal infinitesimals, not only are there infinitely many, but there are absolutely infinitely many, other surreal infinitesimals. The point is, surreal numbers like ORD/2 or infinitesimals resonate with the vague image of the vague threshold between the "amounts"/degrees of generality or particularity required to be abstract or concrete, and mathematically represent (meaning: represent in mathematical style) the way in which the generality-particularity ordering itself is infinitely ascending and descending.

There are other manifestations of this "phenomenon," such as the seeming implexion of numbers like ω - 1 in ω proper. Even an irrational number is a function from/involving a negative hyperoperation, inasmuch as fractions represent division, and an irrational number is identified in terms of an endless such division. Generally, if we mirror the theory of addition's relationship with multiplication, and so on, upwardly, now though in terms of descending hyperoperations, we get all the "weird-looking" surreal numbers, like the square root (or the cube root, or...) of ω, or ω + 3.14..., and so on and on.

With respect to implexion specifically, the application of this concept, by Meinong, to the question of generality and particularity (of how general objects are true of particular ones), indicates that there is some possible merit in a surreal mathematization of the generality-particularity ordering as such. Consider again ω - 1. Now it is true that ω - (ω - 1) = 1, and indeed, we end up with a "premirror" of the mirror of the entire ascending and descending universes, here, viz. ω/2. But to say, "An object crosses the abstraction threshold when the number assigned to its generality is [some weird surreal number]," would not remove us from the realm of vagueness completely; rather, to recapitulate the epistemicist view, we expect that meaningful "computations" of such threshold numbers are outside of human reason, in the sense that even if vaguely possible such computations are admitted and analyzed in broad terms, specific assignments of specific weird surreal numbers to objects, so as to decide the question of their abstraction, are probably not justifiable by applying known/justified mathematical techniques.

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  • General/particular is, in principle, a conceptually different axis from concrete/abstract. People do talk about abstract particulars (Aristotle) and concrete universals (Hegel), albeit controversially. What you discuss seems to be more about general/particular, which non-controversially comes in grades, than concrete/abstract. The latter distinction is between having/not having causal powers, and seems much crisper, although still with plenty of acknowledged borderline cases.
    – Conifold
    Commented Jan 8, 2022 at 5:07
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    Numbers, and mathematics in general, are usually taken as prototypical of abstract entities. Trying to distinguish mathematics from abstractions -- as you do in this question, appears to involve a fundamental misunderstanding of the nature of abstraction.
    – Dcleve
    Commented Jan 8, 2022 at 7:59
  • @Dcleve Are they? Some consider numbers (integers) to be abstract particulars, which would make them non-typical. I would say that groups are more "abstract" than numbers, and categories and motives are more "abstract" still. One can distinguish degrees of "abstraction" within mathematics as well as outside, compare "tree" to "object", although it may be hard to pull apart the senses of "abstraction" and generality.
    – Conifold
    Commented Jan 8, 2022 at 10:16
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    @Conifold, there are orders of abstractions in the sense that there are abstractions of abstractions, but that doesn't suggest that at the low orders of abstraction the objects aren't fully abstract objects as this question suggests. As to numbers being prototypical abstract objects, they were the prime example of abstract objects used by Frege, who first explicated the modern notion of abstract objects. I'd suggest that makes them pretty prototypical regardless of whether they can be predicated of anything. Commented Jan 8, 2022 at 12:37
  • @DavidGudeman "We should be open to the possibility that the best sharpening of it will entail that some objects are neither abstract nor concrete. Holes and shadows, if they exist, do not clearly belong in either category; nor do ghosts, Cartesian minds, fictional characters, immanent universals, or tropes", SEP. I do not see why one cannot extend grades up the chain as well if they please. Frege's treatment of integers sets them apart from how the bulk of abstract objects are typically treated, even by neo-Fregeans.
    – Conifold
    Commented Jan 8, 2022 at 14:29

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