I have a long interest in the 'problem of universals' and a conviction that some form of either Platonic or Aristotelian realism in respect of universals is correct. This extends to what is called 'Platonic realism' in mathematics, i.e. that mathematics describes objects (where the term 'object' signifies for example the natural numbers) which are real independently of the apprehension of any individual. (Contemporary Platonists include Kurt Godel and Roger Penrose.)
Platonic realism is the philosophical position that universals or abstract objects (including number) exist objectively and outside of human minds.
Aristotelian realism holds that universals are incorporeal and universal, but only exist only where they are instantiated; they exist only in things, not in some purportedly ethereal domain or Platonic heaven. For this reason it is sometimes called 'immanent realism'.
I have long entertained the idea that the debate about the reality of universals can be resolved by understanding that universals have a different kind of existence to particulars.
For example, the view of representative nominalists was that only names (or, more generally, words) are universal, "for the things named are every one of them singular and individual" (Hobbes, Leviathan, Ch. 4). According to Ockham universals are terms or signs standing for or referring to individual objects and sets of objects, but they cannot themselves exist. For what exists must be individual, and a universal cannot be that; the mistake of supposing that it could was the fatal contradiction of Platonic realism.
Now, I say that the supposed contradiction in the Platonic/Aristotelian view can be resolved if we understand that universals do not exist in the same way as particulars. Their existence (and whether 'existence' is the correct word in this context is part of the issue) is purely intelligible - but universals are the same for all who think. This is why, for instance, there is universal agreement about fundamental logical laws and arithmetical rules. Regular abstractions such as these provide the basis for language, abstract thought, mathematics, and even science itself.
About one of the only places where I've encountered acknowledgement of this is in Bertrand Russell's discussion in the Problems of Philosophy - The World of Universals, where he says:
We shall find it convenient only to speak of things existing when they are in time, that is to say, when we can point to some time at which they exist (not excluding the possibility of their existing at all times). Thus thoughts and feelings, minds and physical objects exist. But universals do not exist in this sense; we shall say that they subsist or have being, where 'being' is opposed to 'existence' as being timeless.
I believe this confession on Russell's part points to something of extraordinary importance that has been mostly lost to modern philosophy, but which was fundamental to the classical tradition of philosophy.
This is that 'what exists' is, as Russell says, those things which are locatable in time (and presumably space), which I would paraphrase as 'the phenomenal domain' or 'sensable objects'. Due to the overwhelming influence of empiricism in modern culture, this is generally presumed to be the only real domain, whereas the domain of numbers, universal ideas, and the like, is nowadays subjectivised as the products of the mind (hence, 'neural output'). But Russell says they 'subsist' or 'have being' albeit of a different order to the phenomenal. I think the correct term for the kind of being they have is actually 'noumenal' - not in the Kantian sense of being unknowable, but in the original meaning of 'objects of pure intellect' (or nous). So here, we see a restatement of the fundamental distinction between 'phenomenal' and 'noumenal' (or the distinction of reality and appearance associated with classical metaphysics.)
So the question is: (1) do others agree that a valid distinction can be made between the nature of the existence of phenomenal objects, and the nature of the existence of intelligible objects such as numbers and universals? and (2) that this is a distinction that has generally fallen from view in modern philosophy?