As per the SEP article on abstract objects:

Some of the archetypes of abstractness are non-spatiotemporal in a straightforward sense. It makes no sense to ask where the cosine function was last Tuesday. Or if it makes sense to ask, the sensible answer is that it was nowhere. Similarly, for many people, it makes no good sense to ask when the Pythagorean Theorem came to be. Or if it does make sense to ask, the only sensible answer for them is that it has always existed, or perhaps that it does not exist ‘in time’ at all. It is generally assumed that these paradigmatic ‘pure abstracta’ have no non-trivial spatial or temporal properties; that they have no spatial location, and they exist nowhere in particular in time.

Other abstract objects appear to stand in a more interesting relation to spacetime. Consider the game of chess. Some philosophers will say that chess is like a mathematical object, existing nowhere and ‘no when’—either eternally or outside of time altogether. But the most natural view is that chess was invented at a certain time and place (though it may be hard to say exactly where or when); that before it was invented it did not exist at all; that it was imported from India into Persia in the 7th century; that it has changed over the years, and so on. The only reason to resist this natural account is the thought that since chess is clearly an abstract object—it’s not a physical object, after all!—and since abstract objects do not exist in space and time—by definition!—chess must resemble the cosine function in its relation to space and time. And yet one might with equal justice regard the case of chess and other abstract artifacts as counterexamples to the hasty view that abstract objects possess only trivial spatial and temporal properties.

They go on to talk about existing in specific ways in spacetime, some ways counting as abstract, others as concrete. Then they address further issues, etc. So my question is not directly relevant to all that, though it's indirectly relevant somehow, somewhere, I'm sure. Basically, I've "always" wondered, if universals are abstract, and if abstractions are beyond spacetime, then what about universals abstracted from spacetime? Like, if spacetime is "substantival," this might be difficult to imagine doing, since you'd have a seriously individuated entity (the single object of pure spacetime, albeit infinitely large and infinitely divisible in itself) in play; but if spacetime is "relational," then abstracting from the individual multiplicity of relations to the single universal would mean representing an abstract object for the relational property of being spatiotemporal.

So we have a Form of Space and a Form of Time, or maybe just a Form of Spacetime. Of course, Kant's clever move was to agree with some of this picture, but to then as if go on to say, "Ah, but that universal spacetime form you're speaking of... It's not outside of itself, for starters, it's inside itself if anything (or the preceding question, whether spacetime is inside of itself or outside of itself, is a transcendental illusion). Or rather, it is 'inside of' us. It's a form that our own consciousness exists as. And so finally, moreover, that one pure, infinite object you thought about before, the unity of physical spacetime, that just is the very form of spacetime, then."

I am tempted to draw a possible analogy, here, between David Lewis' modal realism, and a picture of abstract vs. concrete worlds as an indexical matter. So just as Lewis made the (bizarre, but intriguing) decision to cast actuality in an indexical light, so we might say that whether an object counts as abstract or concrete depends indexically on which spacetime that object is in: the Formal example, or the physical one? I.e., then, what we call abstract objects from our frame of reference are concrete in the abstract spacetime that they are, in fact, located within (so the cosine function is somewhere, after all); and from their "point of view," it is we who are abstractions. Then things like chess are just especially sharper examples of how, when we switch our own point of view with "theirs" in our imagination, we can make out anything to be abstract or concrete modulo the alternations over the two worlds.

So why wouldn't abstract objects, even if not existent in our spacetime, not have locations relative to a frame of reference constituted out of the abstract objects for the properties/universals of spatiotemporal qualification?

By the way, I've really always thought that the SEP article quoted in the above, was presenting the case for, "The cosine function is nowhere at most," way too strongly, because I could easily imagine saying that the answer is actually that the cosine function is everywhere, since trigonometric functions can be seen to affect the mathematical determinations going on across the entire known universe, all the time. Now, I'm not really trying to prove this statement about the cosine function; to be sure, I'm too mentally exhausted from trying to understand set theory to overcommit to an answer to a random question about physical embodiments of triangle theory. But it seems like an option, at any rate, and curiously overlooked in the quoted article.

  • For the same reason abstract objects are not colored with an abstract version of color. Spacetime as a concept no more has spatiotemporal characteristics and can serve as an abstract container than water as a concept is wet and can serve to quench abstract thirst. In their turn, trigonometric functions cannot be seen affecting determinations of anything anywhere, all that can be seen is us describing some determinations in their terms. It reminds me Hume's famous point that the "causing" is never seen or heard either, we attach it with our own mental hands afterwards.
    – Conifold
    Sep 19, 2022 at 0:36
  • @Conifold, what about tropes? Is a redness trope red? If tropes are abstract particulars, then is there an abstract redness that is itself red? The SEP trope article quotes one source as saying, "... each of these concrete particulars has a red trope—call these tropes F, G, and H—and these concrete particulars exactly resemble each other in colour because F, G, and H exactly resemble each other in colour." Plato thought Forms self-participated in some way. So how would I know that abstract objects are not colored with an abstract version of color? Sep 19, 2022 at 0:43
  • @KristianBerry Asking if abstract objects should have coordinates is a category error. en.wikipedia.org/wiki/Category_mistake Purge your question of the several such errors you have made and re-ask it.
    – BillOnne
    Sep 19, 2022 at 0:58
  • "Red is not a property the trope has; rather, the red trope is the red itself", as IEP puts it. Combine that with SEP's "tropes are abstract because they “(ordinarily) occur in conjunction with many other instances of qualities””, which is not what "abstract" normally means at all, "quite a few trope proponents characterize the trope as concrete", and their general use for nominalistic elimination of universals altogether, to see that they are not very useful for your designs either.
    – Conifold
    Sep 19, 2022 at 1:07
  • @BillOnne, I am not naive enough to believe in category mistakes, ontological commitments, etc., at least not without specific reasons for the specific case. There is no consensus about what abstract objects are, how they are, how we would possibly know about them, etc. Hell, Zalta has a system with contingently abstract objects that become instantiated particulars at times. Sep 19, 2022 at 1:42


You must log in to answer this question.