Mathematicians

The opinions of mathematicians on this matter are varied. While some in applied mathematics feel that they are scientists, those in pure mathematics often feel that they are working in an area more akin to logic and that they are, hence, fundamentally philosophers. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels.

Experimental mathematics

Experimental mathematics is sometimes said to mean the application of the experimental part of the scientific method to mathematics, where mathematicians develop hypotheses before attempting proofs, and then see if their calculations are consistent or inconsistent with their hypotheses. An inconsistency effectively disproves an hypothesis, by providing a counterexample; consistency suggests that it is worthwhile to attempt to prove the hypothesis rigorously.
Although chaos theory and fractals led to an increased emphasis on experimental mathematics beginning around the 1970s, following the work of Edward Lorenz and Benoit Mandelbrot, mathematicians have always done this, and so this is nothing new. Thus experimental mathematics is used in common parlance among mathematicians to refer to a special kind of experimentation, using computers to investigate a large number of cases, or perform computations that are difficult to do by hand. It is fair to say that the use of computers in this manner (not to be confused with automated theorem proving) has become more accepted over time by the mathematical community as a worthy Endeavour. Indeed, some well-respected journals have begun accepting papers that are largely consisting of experimental mathematics, and there is even a journal devoted entirely to it.

Scientific method

Scientific method refers to a body of techniques for the investigation of phenomena and the acquisition of new knowledge of the natural world, as well as the correction and integration of previous knowledge, based on observable, empirical, measurable evidence, and subject to laws of reasoning. Although specialized procedures vary from one field of inquiry to another, there are identifiable features that distinguish scientific inquiry from other methods of developing knowledge. Specific hypotheses are formed to propose explanations for natural phenomena and experimental studies test the predictions for accuracy in order to make increasingly dependable predictions of future results. Hypotheses in a given field of inquiry are logically bound together by a wider theory that assists researchers in forming new hypotheses, as well as in placing groups of specific hypotheses into a broader context of understanding.
Among other facets shared by the various fields of inquiry is the conviction that the process must be objective so that the scientist does not bias the interpretation of the results or change the results outright. The scientific method also may involve attempts, if possible and appropriate, to achieve control over the factors involved in the area of inquiry, which may in turn be manipulated to test new hypotheses in order to gain further knowledge.