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The person you are replying to qualifies their statement by saying that in mathematics numbers exist independently of any physical existence.

I can go with that. If it's understood to mean that 1+1=2, regardless of context.

The ultimate question is philosophical -- do abstractions exist independent of physical existence? Or to phrase it another way -- is there really a distinction between the physical and non-physical?

I think you mean "a distinction between the ~~physical~~ real and ~~non-physical~~ non-real?" The answer is self-evident. Unless one is a deranged subjectivist who feels that his private imaginings are no different than objective reality.

Karl Popper made an attempt to unravel the phenomena of real things that have no physical existence, and unreal things that have no physical existence by dividing phenomena in three worlds: 1) the physical world 2) the imaginary world that tries accurately to describe relationships in the real world and 3) the imaginary world that does not try to be an accurate reflection of the physical world.

In my humble opinion, much of the unsatisfactory vagueness that always seems to hover about ontological discussions on this subject is alleviated by sharply differentiating these three worlds, as Popper did in his fairly famous 3-worlds paper, which I recommend.

It is, I submit, the tendency to implicitly resolve the middle world (accurate imagination) with either the physical world (raising type conflicts for the formal crowd) or with the purely imaginary world (thereby unfairly yielding the tar brush to banish it from objective existence) that leads these discussions into infertile territory.

I've had a sudden impulse of delight which has led me to suggest a modified platonist position: as a new & improved modern platonist, I subscribe to the theory that world-1 objectively exists, world-3 subjectively exists and world-2 plows a middle ground we have not named, and which is neither entirely objective nor subjective. I'm going to call it "conditionally exists" or interjective. The condition being, of course, that world-1 behavior is close enough to world-2 descriptions to qualify as existing, for whatever purposes we have to hand. Something that cannot be guaranteed to be permanent or completely reliable. As an example--Ptolomaic astronomy is an interjective (world-2) reality of less reliable existence than Einsteinian astronomy. Nobody really thinks epicycles are an essentially explanatory orbital description, even though they accurately describe orbits. And nobody knows if Einsteinian astronomy is the final story. So there's an intractibly subjective element to world-2, even though it is objective in intent.

I call it the toe-in-the water school of platonism.

I think you mean "a distinction between the physical real and non-physical non-real?" The answer is self-evident. Unless one is a deranged subjectivist who feels that his private imaginings are no different than objective reality.

I believe in reality. Maybe an even better way to put it would be -- what is the distinction between reality and consciousness? For instance, the color green is a property of our consciousness, while in reality there is only light of a certain wavelength. The same probably holds true for numbers. If there were no conscious beings in the universe would qualia like the color green, or numbers exist?