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If you ask a mathematician today, many will tell you that mathematics is not a science. Many physicists, chemists, and scientists in other disciplines would say something similar. Mathematicians will emphasize several differences between mathematics and empirical science, from aesthetic differences to the unfalsifiability of mathematics. The biggest distinction seems to be that Mathematicians don't accept the Baconian method of induction, having different standards of proof, and as such, can't reasonably be classified as an empirical science.

On the other hand, the original definition of the term "science", as per its etymology, more or less just means "knowledge". The term began to mean "empirical science" (in the sense of Bacon) later, perhaps in the 17th and 18th centuries, but it seems to me that Mathematics was still implicitly included by use of the phrase "the sciences" until significantly later. This was a source of some debate on our Area 51 proposal, and as Conifold suggested, I'm moving it here so that we can get an authoritative answer.

When did use of the term "the sciences" by mathematicians and scientists stop implicitly including mathematics? I am particularly looking for specific quotes/statements from scientists or mathematicians (preferably in English) addressing the question as to whether Mathematics is one of the sciences, though other forms of evidence are also welcome.

For some evidence, Gauss famously referred to mathematics as "the Queen of the sciences", attributed to him by his biographer Sartorius von Waltershausen (but see the comments below questioning the accuracy of this interpretation and translation). On the other hand, by the 20th century, people like Einstein were saying things like "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." In addition, people like Hardy were emphasizing the aesthetic aspects of mathematics, e.g. in his Apology. So it seems to me like the biggest shift happened sometime in the 19th or 20th century, but it's hard to pinpoint exactly when this change began or what sparked it.

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    $\begingroup$ I have my doubts that Gauß really used the English word science and not the German Wissenschaft, which is much wider in definition, or Naturwissenschaft, which can be roughly translated to science nowadays, but still has a history separate from science. Even if he used science, he may not have been aware of the differences. $\endgroup$ – Wrzlprmft Oct 29 '14 at 12:04
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    $\begingroup$ @Wrzlprmft Even if his statement were in English the metaphor would need context. "If the sciences are the king, math would be the queen." "Math is the Queen, Observation/Reason/Scrutiny is the King. Together they rule over the sciences." By itself the phrase does not show the intent of his statement enough to say "Gauss thought Math was a Science". $\endgroup$ – kaine Oct 29 '14 at 14:07
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    $\begingroup$ It would certainly be a reasonable explanation that Gauss' intent was partially changed in translation, and would invalidate part of my evidence, but I don't feel that would completely answer this question. In particular, if one wants to claim that the split had already happened by his time, I'd want to see writings of some of his contemporaries or predecessors opining that mathematics was not to be considered as one of the sciences. $\endgroup$ – Logan M Oct 29 '14 at 16:07
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    $\begingroup$ “but I don't feel that would completely answer this question” – That’s why it’s only a comment. $\endgroup$ – Wrzlprmft Oct 29 '14 at 21:39
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    $\begingroup$ @Wrzlprmft Fair enough, I just wanted to make sure we're on the same page with regards to what I'm asking. $\endgroup$ – Logan M Oct 29 '14 at 21:43
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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.


Summary

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.

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Some people would still call mathematics a science. (V. I. Arnold is a notable example). The distinction became commonly accepted in the first half of the 20-s century, but the process was slow and it was different in different cultures. For example, in the Soviet universities the degree in mathematics is still called "Doctor of physical and mathematical sciences". In the middle of 20-s century there were very few mathematics departments in Soviet universities. Most departments were called "Department of mathematics and Mechanics", and earlier they were departments of mathematics and physics.

If you go further back to 19 century, you discover that many of the greatest mathematicians worked in both mathematics and physics or astronomy. (Gauss, Riemann, Green, Kelvin, for example).

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By the etymology of science, any pursuit of knowledge may be wrongly attributed as science. But science in reality is much more than the just knowledge, it means systematizing the existing knowledge and then suing that knowledge to gain more knowledge and so on ad infinitum. But how do we gain this knowledge, we do that by using what is now called the scientific method, although at times it becomes very difficult to really tell where the boundaries of this method ends.

Mathematics is slightly different, to be called a scientific method we must have some empirical and measurable evidence of what we are discussing. At times, this becomes very hard to achieve. Case in point are statements for which we know we can never give a definite answer. Another instance is the construction of the standard topological set $S_\Omega$, we know that it exists. but we do not know what it is. This in some sense is against the ethics of what we call science.

This demarcation of math and science has become more prominent in the 20th century after many abstract mathematical concepts came into being, which at first sight seemed to be counter intuitive or which lacked any evidence whatsoever about their existence.

But still, we still call it the mathematical sciences when we include fields which are inspired by mathematics, but maybe whose specialists we do not call mathematicians.

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So we are using Wikipedia? If we consult Branches of Science we will find Mathematics listed in there.

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On the other hand, the original definition of the term "science", as per its etymology, more or less just means "knowledge". The term began to mean "empirical science" (in the sense of Bacon) later, perhaps in the 17th and 18th centuries, but it seems to me that Mathematics was still implicitly included by use of the phrase "the sciences" until significantly later. This was a source of some debate on our Area 51 proposal, and as Conifold suggested, I'm moving it here so that we can get an authoritative answer.

English was not yet the dominant language of science in the 17th and 18th centuries. The changed meaning of the term "science" only became important after English had advanced to be the dominant language of science during the 20th century. This coincides well with the time when the term "the sciences" stopped to implicitly include mathematics.

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