You might not have to think of anything of abstract objects devoid of their properties if you give weight to some sort of bundle theory of objects(REP). While there is a lot of support for neo-Platonic thinking, there are empirical and constructivist accounts that see math objects as nothing more than references to abstractions of physical objects or even other abstractions.
This is a problem in the philosophy of mathematics. According to Michael Dummett in his Frege: Philosophy of Mathematics the founder of analytic philosophy, Frege was engaged in exchanges with Dedekind, Husserl, and others about this very question in regards to what a number is. In German, he was able to capitalize on a divergence in lexemes Anzahl und Nummer where the former might be translated as quantity as in conceptual cardinality as opposed to number which is often used as a synonym for digits in English. Therefore, if mathematicians realize a dichotomy between the general and the specific, what some philosophers refer to as universal and particular, then cardinality is reified by collections of digits, whatever there representation (these are known as graphemes in the philosophy of language). Today, this distinction is generally subsumed by philosophers of language between the dichotomy between syntax and semantics, how something is represented versus what it means. This is kith and kin to, for instance, the difference between a sentence and a proposition. 'I am hungry' and 'Hunger has beset me' are synonyms.
What you ask about is a bit deeper philosophically, because it is an ontological question about what it means to exist, and there's a whole conversation in the canon (relatively) recently among Meinong, Carnap, Quine, and other famous folks that tries to hash it out. Stanford has an introductory article on abstract objects.
My take on it as a constructivist is that references can refer to physical objects, mental objects, or other references. Mental objects are mental representations of various inutitional (psychological) faculties abstracted from instances.
Thus, we abstract from a collection of fishes, say trout, perch, salmon, and bluegill, that are subitized or counted, because we can differentiate inuitively between species as prima facie natural kinds and we can metaphorically contain those species as sets, classes, what have you and reference with a category label. Right? In our childhood, we could have gathered blocks shaped as triangles, squares, circles, and hexagons, counted each, done operations on like sets of collections of shapes. The abstract matrix object, therefore, is just a reference to the process and the syntax we use during this activity to communicate with others. Thus, there is a utility and the linguistic behavior is preserved and used in what Wittgenstein would call a language-game. In a nod to our primary sysadmin, such abilities are rooted in faculties of collective intentionality as described by M. Tomasello.