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In the philosophy of mathematics, mathematical practice is used to distinguish the working practices of professional mathematicians (eg. 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. Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will...
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...
In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ...
Peer review (known as refereeing in some academic fields) is a scholarly process used in the publication of manuscripts and in the awarding of funding for research. ...
In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ...
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...
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. Quasi-empiricism in mathematics is the movement in the philosophy of mathematics to reject as pointless the foundations problem in mathematics, and re-focus philosophers on mathematical practice itself, in particular relations with physics and social sciences. ...
The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...
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. As the term is understood by mathematicians, folk mathematics or mathematical folklore means theorems, definitions, proofs, or mathematical facts or techniques that circulate among mathematicians by word-of-mouth but have not appeared in print, either in books or in scholarly journals. ...
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. Pierre de Fermat Pierre de Fermat (August 20, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, southern France, and a mathematician who is given credit for the development of modern calculus. ...
Pythagoras (582 BC – 496 BC, Greek: ΠυθαγόÏας) was an Ionian mathematician and philosopher, known best for formulating the Pythagorean theorem. ...
Euclid of Alexandria (Greek: ) (ca. ...
In mathematics, an irrational number is any real number that is not a rational number, i. ...
Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
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. The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...
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