Let's start with everyone's first go-to source: Wikipedia. Right in the long introduction at the top lies the passage
Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences". Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions".
That's a good start. Later on, we find that
Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper.
Ouch. That's a valid point, though. But if we continue on to the next sentence,
However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently."
And in the concluding paragraph of that section,
The opinions of mathematicians on this matter are varied.
This sums up the sense one gets from reading the section: people are pretty divided.
We can deduce that, given the varying time periods during which these mathematicians and scientists worked, that there has been a lot of debate on the matter for centuries, and that debate endures today. Nobody can seem to agree, which could render the issue moot.
Next up is this rather interesting essay, which starts off with, as its abstract,
Mathematics is not a science, but there are grey areas at the fringes.
Interesting. Let's delve in deeper . . . only to find that it is merely opinionated, citing sources but not anyone famous. It does make some interesting points, though:
- "In mathematics, however, the ultimate arbiter of correctness is proof rather than empirical evidence." This alone seems to separate it from the sciences, which require absolute evidence (or as close to it as possible) for an idea to be accepted. Also (a point of my own), you can never prove a scientific theory; this is clearly not the case in mathematics, although there are some basic axioms that can never be proved.
- "The basic problem is that one can be confident of a fact derived by mathematical methods only to the extent that the mathematical object being considered is an accurate model of the relevant parts of the universe." In other words, many conclusions derived from mathematical models can be proven to be true, though the models are only as accurate as the data they are given.
This essay, unfortunately, merely gives arguments, instead of citing famous mathematicians, and so we will push it aside. I'd also suggest this page to look at criteria for determining what a science is.
This last bit is a personal opinion, so feel free to ignore it.
I get the feeling that mathematics began to separate from the sciences when it became more abstract. With the rise of pure mathematics, many mathematicians began to venture into the discipline entirely for the sake of mathematics, without any thought for its applications to physical theories. We could put a finger on this point at some time during the career of David Hilbert, who, while making extraordinary contributions to applied mathematics, also made numerous advances in pure mathematics. In fact, Wikipedia credits him as heavily influencing the field of pure mathematics:
At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof.
I love getting Bertrand Russell in there, but I would argue that Hilbert's devotion solely to mathematics (as opposed to Russell, who could be viewed as a jack-of-all-trades) puts him at the top of the list of those leading the charge into pure mathematics, thus taking it away from the sciences.
The debate about whether or not mathematics is a science still goes on today. The center of the debate lies on the empirical predictions (or lack thereof) or purely mathematical ideas, and whether or not mathematical ideas that can be proved to be true can be true in the real world.