Some examples of mathematical social constructs for ariels:

In our everyday math, division by zero is "undefined". But there's nothing to prevent someone from defining it, and if they try, their math will come out differently:

Heracles has six cows. He divides them among four children; they get each one cow, and two cows remain to him. He divides them among three children; they get two cows each, and none remain. He divides them among two children; they get three cows each, and none remain. He divides them to one child; that child gets six cows, and none remain. When he divides them to nobody, six cows remain to him.

Try and tell Heracles he can't keep his cows.

Similarly, we've invented something called a negative number. A society could easily create a mathematical system where such a concept is "undefined".

Torddjen has seven phasers. The system computer MegaHAL informs him that Ko'lei has a claim of two phasers on him, Abraxas four, and !Kou nine. Torddjen knows that Abraxas's cyborg thugs are very unforgiving, so Torddjen gives Abraxas his four phasers. Ko'lei is his friend and won't fault him if he delays another week, so he gives his remaining three phasers to !Kou. MegaHAL does not say that now Torddjen has "-8 phasers". That is meaningless. Every debt must have a debtor.

See if you can explain to MegaHAL who exactly Torddjen has -8 phasers in relation to, without getting both you and him thrown out the pod bay doors for doing math backwards.

I conclude that some math is a social construct.

"As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality." --Albert Einstein

ariels: The Torddjen example isn't mine, it's an adaptation of an example that came up in a linguistics mailing list I'm on in a discussion of number systems in our languages.

"Division" for Heracles means similar to what it does for us, i.e., the meting out of a quantity of objects into an equal number of groups. The difference between our "division" and Heracles's is the question of the remainder--for us, it is divided equally among the groups (5obj / 2grp = 2.5obj/grp) in which case dividing into zero groups is quite meaningless, but for Heracles it's simply not meted out at all (5obj / 2grp = 2obj/grp, R1) in which case dividing into zero groups is not meaningless. You are right that it is not the same as real numbers. The social construct here is that our society uses real numbers, and his does not (regardless of whatever consistency or truth-value there is in the systems themselves).

You are right that natural language is ambiguous. I think the ambiguity here is that I'm applying a different definition of math to the thesis of this node than you are. You might say I mean the science of mathematics is a social construct, and you mean that the mathematics themselves are not, and I would have no problem with that.

an update, after thinking about it more (leaving behind the original because people replied to it)

If you take a planet and put another planet next to it, you have two planets. This is only true because our language (a social construct and the basis for all social constructs) creates a category ("planets") that encompasses both of them.

You cannot count things until you categorize them. (or, you can't enumerate the contents of a set until you define the set.) And this categorization, even if it is so broad a category as "thing", requires a categorizer.

We start with categories. We begin to count. We begin to take generalizations about the real world and develop them as abstractions and axioms. We choose ideal, clean situations: we believe in the commutative property of addition, for example, because we have abstracted the idea of grouping identical, discrete objects that have no effect on each other.

Once we have developed our axioms, we construct our math on top of that. We don't choose the math (which is what saying "math is a social construct" sounds like) we simply discover the complex system that our simple foundational assumptions imply.