Platonic realism is a commonly held philosophical position about abstract ideas, but it is very hard to argue for it being coherent.
Abstract ideas are, in my humblest of opinions, the source of the most important arguments in philosophy. The ontology (the nature of the existence) of abstract objects underwrites all other philosophy from morality to mathematics, aesthetics to personhood. You can’t make serious claims about these latter ideas without first understanding the impact of the former.
Much of my writing (both here at OnlySky and in my books) has espoused my position of conceptual nominalism—in short, the theory that all abstract ideas are concepts that do not exist independent of our minds. If all sentient life were to die out, then there would be no morality, no mathematics, no human rights. We construct these ideas after arguing among ourselves until we agree.
Agreement in these matters usually ends up with dictionaries and encyclopedias being written to reflect the consensus understanding of words and ideas, theories and concepts. These are then manifested in laws that are enacted by lawmakers we vote into positions of power. They become meaningful when they are enforced by entities such as governments, police forces, and legal teams.
All this is shorthand for saying that morality is constructed. Morality, like all abstract ideas, exists in here (your minds and mine) and not out there, in the aether.
On the other hand, those philosophers who are not nominalists of sorts are realists, and often espouse some kind of Platonic realism. The Greek philosopher Plato believed that abstract ideas such as universals exist outside of human minds, with no spatiotemporal dimension. He espoused some realm where the perfect or ideal form of all ideas existed. Our understanding of such is akin to seeing shadows of these perfect forms on a cave wall rather than seeing them first-hand in clear sunlight.
In other words, mathematics of morality exist irrespective as to whether we do or not, and humanity merely discovers it.
The problem with Platonism is working out what effect it really has (with many of the following points coming out of an email conversation with Richard Carrier). There are two horns to a dilemma: either Platonism asserts nothing (such that there is no distinction between it being true or false) or it asserts something. Yet if it asserts something, it asserts that abstracts exist and that they have some sort of effect, but that they simultaneously “never exist” and “exist nowhere” (given they have no spatiotemporal dimension).
This appears to be somewhat incoherent.
There is a pragmatic element to this. This is similar to the position of verificationism: “the philosophical doctrine which maintains that only statements that are empirically verifiable (i.e. verifiable through the senses) are cognitively meaningful, or else they are truths of logic (tautologies).”
If we cannot detect abstracts in that they have no causal efficacy (though we construct conceptual understanding of them in our minds), then there is no difference to the world between Platonism being true and Platonism being false. This then renders the word “Platonism” vacuous and devoid of content.
On the other hand, if there is a difference between a world where Platonism is true and one where it is false, then that difference by definition has effects because a world where it is true will be changed with respect to a world where it is false. Those effects must have a location; otherwise, they don’t exist. If the effect of Platonism being true exists at no time and in no place, then those effects functionally don’t exist.
Or Platonism doesn’t matter.
The job of the Platonist is to explain how abstracts have those effects (how they make the world different from a world where Platonism is false). They must also explain how such entities can have effects at particular times and places, without those entities ever existing in any of those times and places.
Claiming Platonism can be true without answering any of this amounts to saying Platonism asserts nothing and is devoid of meaningful content.
The problem here is perhaps an epistemological one whereby if we cannot know the Platonic reality—even if it does somehow exist—or if we cannot sense it in any way, then it might as well not exist. It carries no pragmatic utility. This is rather like divine command theory, where if we cannot know which god and which commands are true, we are left constructing morality ourselves anyway.
If we have no idea how well we are reflecting or accessing this Platonic realm, then there is no real meaning or impact to that realm.
This is all very important because if one’s moral value system is deontological—a rule-based objective system—and one which depends on some kind of Platonic realm where the abstract rules reside, then the argument over abstracts has some serious ramifications.
It is always interesting to cogitate on what the state of affairs would be if all sentient life were to die. In the absence of conceivers, there would simply be no concepts.