It’s just because of the way I defined truth. Do you have a different definition? My definition of truth is the correspondence of the mind with reality. I can make up or imagine something that is not real, and my mind can have a perfect grasp of it, but that perfect knowledge will not fit my definition of truth. The definition I have given is the philosophical definition of truth, according to which truth and being are interchangeable. We are here talking about the philosophy of mathematics. So, I stick to the philosophical definition, which requires that truth be related to being.
Yes, it is a stretch. I can say that the proved statements are valid or correct. But I can’t say they are true. You can, of course, redefine truth to mean anything that is consistent, valid, or correct. In that case, you can say that all consistent mathematics, including those that have no bearing with reality, are “true.” But that is not the philosophical definition of truth.
The “sphere” is the conceptualized property of actual spherical objects which may not be perfectly spherical. In a way they are like the essences of things. When you see a child that is born with a defect, don’t we still say that the child is human? The essence of humanity still exists in the child even when the child has an imperfection. The essence “sphere” exists even in objects that are not perfectly spherical, otherwise the mind will not be able to abstract and form a concept of it. The real mathematical objects that I have been talking about are those that were derived from real things, not those that were derived from other concepts. Spheres, cylinders, lines, triangles – all these are real mathematical objects, whose concepts were derived from real things that are not perfect.
But you can also have concepts that are derived, not from real things, but from other concepts. For example, you can have a concept of a “four-dimensional surface” derived from your concept of a three-dimensional surface. Such “four-dimensional surfaces” are no longer real. Statements that we make of them may still be judged as valid or invalid, correct or incorrect. But from a philosophical standpoint, they are neither true nor false, unless you redefine the meaning of truth.