What do contemporary philosophers mean when they say abstract objects exist (or realism of abstract objects)?

Question

So for Plato, forms exist in some other "divine" realm similar to the way concrete objects exist in our ordinary world. So the existence of forms is comparable to the existence of concrete objects... except not in the same realm.

But what exactly do contemporary philosophers mean when they say abstract objects exist. I doubt they mean they exist in some other dimension as perfect versions of the ones we have in our world.

Eg: Putnam and Quine are called mathematical platonists:

https://en.wikipedia.org/wiki/Quine%E2%80%93Putnam_indispensability_argument

They believe abstract mathematical objects exist... but what exactly does this mean? I really don't understand what exist means other than the way concrete objects exist, except in some other realm/dimension. Is that what they mean by existence of mathematical objects?

So what are nominalists and realists actually debating about when they say abstract objects don't exist vs exist?

Answer 1

I don't think that Putnam or Quine (or for that matter Plato) were all that concerned about the exact nature of these 'abstract objects'. The point is that they felt certain other philosophical principles had to be satisfied, and so numbers had to exist as objects for those principles to hold. I tend to blame this modern excursion into idealism on Analytic Philosophy. Analytic Philosophy held that proper language must be denotative — that words must point to objects or relations in a systematic and robust manner — and therefore numbers had to be objects of some sort, otherwise the philosophical worldview deteriorates.

The problem is exacerbated by Anglophone empiricism, which is actively anti-subjectivist. Empiricism rejects the idea that 'figments of the mind' can be construed as objective entities; it focuses on 'empirical' (observable/measurable) experiences explicitly to exclude faith and mere belief from being entered as authoritative. European theorists wouldn't have as much of a problem here. They would likely treat numbers as a special kind of social fact whose existence is ordained and enacted by social institutions and pressures (i.e., even the primitive urge to say "these are my cows and those are yours" would require an 'accounting' that would invariable demand numbers). The numerical platonism/nominalism dispute is kind of a tempest in a teapot, having relevance to a number of academics in certain departments of Anglophone institutions, and not affecting much outside that group. It's a fine way to spend one's time if one likes that kind of thing, but it smacks of knights jousting for the honor of a lady's hand while the lady has gone and gotten on with her life.