Dear @Sudev, scientific models could not be expressed without mathematics. I would say there is no science without mathematics. In some science subjects the proof can be confirmed exactly (applied research)! "Whether mathematics itself is properly classified as science has been a matter of some debate. Some thinkers see mathematicians as scientists, regarding physical experiments as inessential or mathematical proofs as equivalent to experiments. Others do not see mathematics as a science, since it does not require an experimental test of its theories and hypotheses. Mathematical theorems and formulas are obtained by logical derivations which presume axiomatic systems, rather than the combination of empirical observation and logical reasoning that has come to be known as the scientific method. In general, mathematics is classified as formal science, while natural and social sciences are classified as empirical sciences."

http://en.wikipedia.org/wiki/Science

I remember Einstein has a comment on why he chose Physics over Mathematics. His main point, if I remember correctly, is that Mathematics is too diverse while the objective of Physics is much more clear. I somehow agree.

Dear @Sudev, scientific models could not be expressed without mathematics. I would say there is no science without mathematics. In some science subjects the proof can be confirmed exactly (applied research)! "Whether mathematics itself is properly classified as science has been a matter of some debate. Some thinkers see mathematicians as scientists, regarding physical experiments as inessential or mathematical proofs as equivalent to experiments. Others do not see mathematics as a science, since it does not require an experimental test of its theories and hypotheses. Mathematical theorems and formulas are obtained by logical derivations which presume axiomatic systems, rather than the combination of empirical observation and logical reasoning that has come to be known as the scientific method. In general, mathematics is classified as formal science, while natural and social sciences are classified as empirical sciences."

http://en.wikipedia.org/wiki/Science

Mathematical & scientific models may be used in other fields, but qualitative models/researches of social sciences may not perfectly find place in mathematics.

1) Mathematical research tends to have one or two authors, while research in most sciences frequently has dozens (that's also why there's no "lead author" in mathematics journals).

2) The use of experiments in mathematical research is more often extremely worrisome for large numbers of mathematicians (I can't recall which proof it was but there was a proof that relied on computer simulations or something similar, and I'm sure mathematicians are still arguing over whether this can be proof).

3) There is mathematical research and then there is mathematical research proper (i.e., papers published by those with doctorates in mathematics and in journals that are mathematical journals as opposed to e.g., papers in econometrics journals, psychometrics journals, or other journals that are for mathematical papers relating to a particular scientific field or discipline). A great discussion I was involved in here a while ago concerned fuzzy statistics and a statistician was asking why statisticians seem to be so critical of fuzzy statistics and the publication in journals like Biometrika reflect this. I found this curious, because I've read a great deal of papers from mathematical/statistical journals that, far from being critical of fuzzy statistics, added new methods for those who use fuzzy set theory, fuzzy probability, etc.

In the end, we both agreed that it was likely that while those whose work is in statistics in general (and may have degrees in engineering, psychology, economics, mathematical physics, applied mathematics, etc.) are a larger category and include a lot more researchers who care about what works, statisticians whose actual doctoral program and post-doctoral research is in statistics are far more likely to care about theory.

Basically, there are a lot of people whose specialty is in one or more mathematical fields, but mostly as work in said fields concerns a specific scientific field or scientific discipline. Then there are mathematicians. To use just one more (personal) example, I grew up down the street from a professor of mathematics at Brown (and was friends with his son). Early on in my work on complex/dynamical systems and nonlinear dynamics I went to talk to him, as he had written one of the graduate texts I used as well as published papers on dynamical systems. While our discussion was extremely valuable and helpful for me, it wasn't in the ways I had expected, mainly because even though we were both talking about the same subject he was approaching it through "pure" mathematics: non-Archimedian properties, dynamical degrees as measures of geometric complexity of iterates, Zariski density orbits of Mordell-Lang linear dynamical sub-varieties, etc. Later, more of his work became more relevant as I was able to understand how it related to dynamical systems from a modelling or empirical perspective. But most of his work is unusable for me because it concerns technicalities with little to no relevance to nonlinear dynamical systems/complex systems research in general.

4) Mathematical research is easier to publish in the sense peer-review is much more straightforward: there are usually few references, most of the paper is couched in the language of the particular mathematical field concerned and for potential expert reviewers is therefore much easier to assess as it doesn’t depend on thinking critically about whether the design had flaws or whether certain findings in a related field are relevant enough to require addressing or even something as simple as “who is this person and how do I know whether she/he has the relevant qualifications?” After all, the relevant qualifications for publishing mathematical research is being right.

5) Mathematics is perhaps the most contentious discipline when it comes to classifying what is or isn’t a science. It used to be the “queen of the sciences”, but the fact that most of Hilbert’s 23 problems were “solved” by:

a) A proof that the problem is unsolvable (e.g., the combination of Gödel and Cohen’s proofs on the continuum hypothesis)

b) A proof that the desired result was impossible or at least impossible in the way Hilbert really desired (e.g., Gödel’s incompleteness theorem; also, in terms of Hilbert’s program in general, another death knell was delivered by Turing’s famous paper on the Entscheidungsproblem)

c) partial or no proof

combined with the general decline of logical positivism and the incredible growth of new scientific fields seems to have resulted in most people considering mathematics to be a tool of scientists and a non-scientific discipline, rather than a “science”.

6) Unlike applied mathematicians or mathematicians like Penrose whose work is in the sciences, mathematical research tends to involve a lot less use of mathematical software and a lot of it can be done as it has been for centuries: writing stuff down on paper or another medium.

7) “Something like half of mathematics consists of showing that some space is compact” (Hubbard & Hubbard’s Vector Calculus, Linear Algebra, and Differential Forms, 4th Ed.). Mathematics is not a field the way classical philology, cognitive neuroscience, systems biology, stellar astrophysics, etc., are. It not only has plenty of fields but its own main branches (both the classical divisions that I feel are dated and others that are modern but subjective). There is nothing which “half of” some science consists of that so specific (at least not that I can think of).

8) The sciences have become increasingly interdisciplinary. I like to use the example of HCII conferences, where one may run into a sociologist one moment and a military official the next, where one talk may be about social neuroscience and the next robotics. Increasingly degree programs are emerging that have no department but are interdepartmental. Cognitive science research is filled with work by computer scientists, linguists, philosophers, physicists, neurologists, psychiatrists, neuroscientists, social psychologists, even researchers from managerial science. JR Christy (an infamous critique of anthropogenic global warming) has been involved in all the assessment reports after 2001 (when he was still considered mainstream enough to be a lead author) mostly because there simply aren’t enough people with the requisite expertise, and the other leading scientist here is even worse (Roy Spencer). One of the most ardent proponents of Robert Rosen’s argument that living systems (technically, [(M,R)]-systems) are closed to efficient causation is A. H.  Louie, a mathematical biologist, whose papers critical of opposing arguments and his own supporting works are published in journals like Artificial Life, Chemistry & Biodiversity, and Journal of Cosmology. Musicology includes a large number of mathematical and statistical research, from the two-volume Musimathics: The Mathematical Foundations of Music to proceedings of the international “Mathematics and Computation in Music” conferences. Basically, fields in the sciences are generally quite specialized and research is increasingly involving experts from a number of diverse fields and backgrounds.

Meanwhile, there are still those who continue to think the classic four-fold division of mathematics- Foundations, Geometry, Algebra, and Analysis- are sufficient. Certainly, the growth of new fields in mathematics is nothing at all like that of the sciences. Geometry and algebra pre-date the natural philosophy of Newton, yet the changes within physics, including the growth of new fields and appearance of interdisciplinary fields like biophysics, nanoscience, etc., far, far, far outstrip the comparable growth in mathematics (I realize that Euclid didn’t know anything about the arithmetic of elliptic curves and classical algebra not only consisted of many developments but is itself far outdone by work in modern algebra; however,  I’m comparing the growth of fields of mathematical research themselves to those of physics, not development within a field).

Unlike in science research, mathematics research or pure mathematics research, rarely has a hypothesis to test. Mathematics research rarely use statistical tools, but rather create new concepts and theories which are then proven rather than experimentally tested. The results of mathematics research is a tautology and not just a statistical probability.