Mathematical practice
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In the philosophy of mathematics, mathematical practice is used to distinguish the working practices of professional mathematicians (e.g. selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the final proof can be formalised, and seeking peer review and publication) from the end result of proven and published theorems.
This distinction is considered especially important by adherents of quasi-empiricism in mathematics, which denies the possibility of foundations of mathematics and attempts to refocus attention on the ways in which mathematicians arrive at mathematical statements.
The modern mathematical practices are what distinguish modern professional mathematicians from older ideas of folk mathematics. Although such "folk" practices may well include useful formulae or algorithms, they are generally without the accompanying proof discipline.
The evolution of mathematical practice was slow, and some contributors to modern mathematics did not follow even the practice of their time, e.g. Pierre de Fermat who was infamous for withholding his proofs, but nonetheless had a vast reputation for correct assertions of results. Likewise there is contrast between the practices of Pythagoras and Euclid. While Euclid was the originator of what we now understand as the published geometric proof, Pythagoras created a closed community and suppressed results; he is even said to have drowned a student in a barrel for revealing the existence of irrational numbers. Modern mathematicians admire Euclid's practices, and usually frown on those of both Fermat and Pythagoras. Nonetheless, all three are considered important contributors to mathematics, despite the variance in method.
One motivation to study mathematical practice is that, despite much work in the 20th century, some still feel that the foundations of mathematics remain unclear and ambiguous. One proposed remedy is to shift focus to some degree onto 'what is meant by a proof', and other such questions of method.
If mathematics has been informally used throughout history, in numerous cultures and continents, then it could be argued that "mathematical practice" is the practice, or use, of mathematics in everyday life. One definition of mathematical practice, as described above, is the "working practices of professional mathematicians." However, another definition, more in keeping with the predominant usage of mathematics, is that mathematical practice is the everyday practice, or use, of math. Whether one is estimating the total cost of their groceries, calculating miles per gallon, or figuring out how many minutes on the treadmill that chocolate eclair will require, math as used by most people relies less on proof than on practicality (i. e., does it answer the question?)
