I COME UP WITH THESE GREAT IDEAS AND YOU **** ALL OVER THEM

YOU ****ING BITCH. ALWAYS NAYSAYING EVERYTHING I CREATE.

What is an abstract object?

Personally, I feel driven to accept their existence, largely from considerations relating to the objectivity and necessity of mathematics, yet I can't help but feel they cause a bigger problem for materialism that is usually acknowledged - especially considering the comparative energy that is devoted to issues like the mind-body problem which I feel have more obvious solutions.

Numbers exist, but not as objects. I don't think numbers exist outside of mind; numbers of things exist, but the number itself is an abstraction created only through conceptualization, which is a mind-dependent process. Numbers are only as real as any other idea.

Numbers exist, of course. Of course not as objects, just as measurements of length exist, but there is obviously length.

Some people don't believe they exist, or that there are actual numbers.

I do think mathematics is all in our heads and is in that way subjective, but the truths therein are universal and are in no way subject to whims or personal fancies. If we agree on the terms, we will all independently come to the same conclusions (if we're doing it right!) even though the abstract objects we're dealing with exist only in our subjective consciousnesses. Subjective doesn't always have to mean arbitrary.

Care to elaborate on that, considering the qualifications I tacked onto my use of the term?

The numbers themselves, yes. They do not exist objectively, even though our understanding of them is gained through observation of discrete entities. So, mathematics -- knowledge concerning relationships between numbers -- is a subjective experience even if the laws therein are universal and absolute.