This seems to be an interesting question and needs attention from experts in the concerned field.

Mathematics is the study of abstract structures that are established from fundamental axioms of truth of reality, using logic and reason. It is these properties that enable mathematics to study nature and in most cases it is said to be the language of nature as well. When the abstract variables represent behaviors of natural phenomena, we have what is called applied mathematics. Today applied mathematics and pure mathematics do have a wide overlap, each enables/nourishes  the other to grow in every dimension. Besides mathematics can be seen from not only its application but from its elegance and beauty/aesthetic. I have attached my collection of artistic images at my website and also in the American mathematical society

http://www.dejeniea.com/

http://www.ams.org/mathimagery/thumbnails.php?album=22

If I read that correctly enough, the one-sentence definition was:

Mathematics is a set of assumptions, its properties and applications.

Some might argue, other fields, such as philosophy, do the same thing!

Somewhere in the definition, the idea of supporting precise quantification needs to appear, I would think. This is what gives math its strength. The precision that it offers, as opposed to vague arguments using just words, that are subject to individual interpretation and biases. I'm not saying that math is only numbers. I'm saying that math has to be able to support precise quantities - numbers.

Math is a different language. It's not uncommon for some concepts to be easier to express in one language than they might be in another. Anything that requires precision, quantification, requires the language of mathematics.

So in short, I would not limit the definition to just explaining the structure of a proof or theorem, though, because to me, those are structures for making an argument. Such arguments may use math, or they may not. Any argument you can name will include assumptions and properties, no?

Design is also a set of assumptions, its properties and applications......