I wrote this piece sometime ago but for some reason never got round to posting it, possibly because I think it really needs expanding. However the years I have spent studying both the philosophy of mathematics and the philosophy of science have convinced me that an expanded version of this answer would become a fairly substantial book; a book I have no real desire to write. I have chosen to post this piece now because the Irish student of philosophy Cathy who inhabits my twitter stream as @Cathyby recently posed the question dealt with here, so Cathy my answer to your query.
I want to turn my attention to a question that has bothered me whenever I have met it in one form or another in the intertubes, “is mathematics a science?” Usually one meets the question in the form of a denial, “but that’s mathematics and that’s not a science”. I have deliberately not chosen this question as title for this post because I don’t think it’s actually a legitimate question as it’s based on a mistaken idea of what science is.
The question that I shall be considering here cannot be asked in German or rather it would be rather strange. In German each academic discipline is a Wissenschaft. Wissen is the German for knowledge and the suffix “shaft” is equivalent to the English suffix “hood” as in neighbourhood or brotherhood and functions as a collective for everything that falls under the concept, so Wissenschaft is everything that falls under the concept knowledge. It is interesting in this context to remind ourselves that both the Latin word scientia and the Greek word mathema also originally meant knowledge. German differentiates between the different types of knowledge so the closest it gets to the common English understanding of the word science is Naturwissenschaft, which however can be translated as the natural sciences and this brings us to what I consider to be the crux of the problem.
All those who vehemently deny the status of science to mathematics have a very limited concept or view of what constitutes science. They believe there is one thing called science that employs something called “the” scientific method. What they actually mean is physics or the physical sciences. Contrary to what these people think there is no monolithic scientific method but rather a fairly large set of related and similar methods that are employed in different branches of the sciences. What constitutes a test for a hypothesis in biology is not necessarily the same as that which constitutes a test for a hypothesis in physics. This is actually tacitly recognised in that we group the sciences according to the subject matter that they investigate and the methodology that they use. We differentiate between the physical sciences, the life sciences, the earth sciences, the social sciences and so on and so forth. Each group delivers its own form of knowledge conform to the subject matter of its investigations.
Mathematics belongs together with informatics (computer science) and formal or symbolic logic to the formal sciences or in German die Formalwissenschaften. These sciences are distinguished by the fact that their true statements are true as a result of their form and not their content, in all three we talk about well-formed statements. Like physics or biology mathematics delivers knowledge and as such is without question a science but it is a different type of science to physics or biology, which in turn also differ from one another.