What is Mathematics About?
Various suggestions have been made about this question: mathematics is about symbols, or mental constructions, or abstract Platonic entities. We can also ask what physics is about and expect a variety of answers: the sensations of the physicist, mind-independent material bodies, an all-pervading consciousness, abstract structure. In the physics case another answer has sometimes been contemplated: physics is about something whose nature we do not know and perhaps cannot know. This usually gets expressed as the thought that matter is an I-know-not-what, a mysterious substratum, a noumenal thingummy. We may know something of its structure and its mode of operation but we don’t know its inner nature. But I don’t know of any analogous view of the subject matter of mathematics: the view that mathematics is about something unknown to us yet partially described by our mathematical theories. Arithmetic, say, aptly represents the structure of mathematical reality, but nothing in it provides a clue about what numbers really are; nor do we have access to anything else that informs us of the nature of number. Thus we have agnostic realism about the mathematical world: numbers are real but we must be agnostic about the intrinsic character of numbers—as we must be agnostic about the true nature of what we call “matter”. Maybe physics and mathematics are ultimately about the same thing, but if so we are ignorant of what that thing is. The advantage of this way of thinking, in both areas, is that it allows us to avoid being forced into unpalatable positions: none of the standard positions is free of difficulty, and at least the agnostic realist position avoids these difficulties. Certainly the long history of mathematics gives the impression of people stabbing in the dark unaware of the vast mathematical world that would later be revealed; the very idea that mathematics has a subject matter would be alien to these early thinkers. The reason is simply that we are not faced with any such subject matter by our senses or by anything else. There is a subject matter to mathematics, objectively real and determinate, but we have next to no knowledge of its ultimate nature; we don’t grasp the underlying mathematical reality (that very concept may be inadequate to its intended referent). The numerals we use are just symbols for we-know-not-what, mere placeholders. That, at any rate, sounds like an option to be added to the usual options. Call it mathematical agnosticism.
 See for example Dirk. J. Struik, A Concise History of Mathematics (1987), which begins 10,000 years ago. Perhaps the earliest mathematicians would say that its subject matter consists of cows and corn, friends and foes.