Not sure that I agree as does a mathematical circle actually exist? We can produce things that approach the concept of a circle and similarly, we can measure circumference and diameter to a level of precision to approach the value of pi, but we never have a perfect circle or the exact value of pi.
I tend towards maths being distinct from physics as some areas of maths deal with concepts that can only have a passing resemblance to reality - the Banach-Tarski paradox is an example. (Similarly, pretty much any treatment of infinities ends up to have little relation to reality such as with Hilbert's Hotel).
I think you're using a different meaning of "exists" than I am.
Ideas don't exist as you can't point at them, steal them, destroy them etc. I can point at something that approaches the concept of a circle and I can point at a set of objects that can be counted, but I can never see a mathematical circle (zero thickness would make it impossible to see) and I can't see a "four" without representing it by a symbol or collection of objects.
I guess one way to decide on a definition of ‘exists’ — since a given thing probably either exists or doesn’t — is to instead decide when it doesn’t exist.
Does a five-sided triangle exist? Well, the very concept seems to be self-contradictory, but surely the idea exists (since I just mentioned it, and you probably understood what it meant and could see why it was contradictory), and even if you don’t believe the idea exists you must eventually concede that the sentence that describes it exists!
If you think things can only exist if you can point at them, steal them and destroy them, then would you say the natural numbers (= 0, 1, 2, 3, …) don’t exist? That would seem to be a strange notion of existence.
Other than that it isn’t a physical object you can hold in your hand, I don’t see why there’s anything problematic about regarding the circle as an object that exists, even if it’s just (in some sense) a theoretical object.
If you take your position to its extreme (but, I would argue, logical) conclusion, almost nothing in your everyday life exists. It’s all atoms, and the fact that you regard, say, a guitar as being a guitar is rather weird because any given guitar is almost certain to be atomically distinct from every other guitar that has ever existed. Are guitars just ideas? Would you say there’s ‘no such thing as a guitar’ because they’re all just approximations to the ideal guitar?
Put simply: what does rejecting the existence of ideas buy you? I don’t see how it’s a productive (or very insightful) position to hold.
I'm simply drawing a distinction between something that physically exists versus something that doesn't physically exist.
Ideas don't (physically) exist although a representation of an idea can exist e.g. a mathematical circle doesn't physically exist, but representations of circles do exist and they vary with how close they get to the idea.
With your guitar example, you demonstrate how we classify things in a messy way. Because the idea of a guitar doesn't exist, but lots of representations of guitars exist, we just refer to those objects as "guitars" as we can recognise the "guitar idea" that they are made to represent.
Distinguishing between physical existence and non-physical existence is very useful - it allows you to separate the map from the territory. It's very useful to abstract the concept of "five" and recognise that it's different to five objects, despite them being so closely related. The abstraction of numbers becomes more useful when considering ideas such as negative or imaginary numbers - it would be difficult to reason about and use them if we continually think of numbers as only "counting objects".
I tend towards maths being distinct from physics as some areas of maths deal with concepts that can only have a passing resemblance to reality - the Banach-Tarski paradox is an example. (Similarly, pretty much any treatment of infinities ends up to have little relation to reality such as with Hilbert's Hotel).