Math most certainly is a social construct.
The utilitarian look on math is that we use it to break down larger things into smaller, easier to grasp, pieces. This reflects our own nature as human being, as we look in on this world through a limited set of perceptions, viewing (or arguably projecting). More importantly, our perceptions, the concept of an object being separate from another object in one's mind, is needed only because of our inability to see everything, the universe and even more, as one single whole.
The example of putting two planets next to each other and insisting that a planet plus a planet must be two planets is flawed because of the word, "two." By specifying that anything has a number greater than one, you're identifying separate objects. The only reason we can distinguish between this and that is the advanced pattern matching software within our brain. This is explained in an excellent fashion by Alan Watts in The Book: On the Taboo Against Knowing Who You Are1; Watts shows a picture of the universe as it is, a seemingly arbitrarily chosen curvy shape, and then shows how humans see the world, the same curly shape with a grid over it.
One may argue that the number one, or two, or three, etc. in pure mathematics has no such connected value or associated material pattern, and is 100% abstract. Yet even then, the numbers are only a reflection of electrical impulses shifting within our brain, much as a computer shifts bits back and forth as it calculates. The only real number, the only true math, is the whole:
1
1In the Vintage Books Edition 1973 edition, this illustration can be seen on page 53.
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