We do? In my experience mathematicians almost always assume base 10 and don't bother indicating it.

In fact we mostly indicate the base when there is a good possibility that it isn't 10, and any stray numbers fail to have a base represented, they should be assumed to be base 10.

(Mathematical notation mostly uses sensible defaults except for differential geometry.)

Yes – but if we do want to talk about other bases, we use this format. Implicit default for a number x is obviously (x)_10, as mathematicians are lazy.