Mathematical practice comprises the working practices of professional mathematicians: selecting theorems to prove, using informal notations to persuade themselves and others that various steps in the final proof are convincing, and seeking peer review and publication, as opposed to the end result of proven and published theorems.
Philip Kitcher has proposed a more formal definition of a mathematical practice, as a quintuple. His intention was primarily to document mathematical practice through its historical changes.[2]
The evolution of mathematical practice was slow, and some contributors to modern mathematics did not follow even the practice of their time. For example, Pierre de Fermat was infamous for withholding his proofs, but nonetheless had a vast reputation for correct assertions of results.
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 éclair will require, math in everyday life relies on practicality (i.e., does it answer the question?) rather than formal proof.
Textbooks or lecture notes which display the mathematical material to be covered/taught within the context of the teaching of mathematics. This requires that the mathematical content being taught at the (say) undergraduate level is of a well documented and widely accepted nature that has been unanimously verified as being correct and meaningful within a mathematical context.
Workbooks. Usually, in order to ensure that students have an opportunity to learn and test the material that they have learnt, workbooks or question papers enable mathematical understanding to be tested. It is not unknown for exam papers to draw upon questions from such test papers, or to require prerequisite knowledge of such test papers for mathematical progression.
Exam papers and standardised (and preferably apolitical) testing methods. Often, within countries such as the US, the UK (and, in all likelihood, China) there are standardised qualifications, examinations and workbooks that form the concrete teaching materials needed for secondary-school and pre-university courses (for example, within the UK, all students are required to sit or take Scottish Highers/Advanced Highers, A-levels or their equivalent in order to ensure that a certain minimal level of mathematical competence in a wide variety of topics has been obtained). Note, however, that at the undergraduate, post-graduate and doctoral levels within these countries, there need not be any standardised process via which mathematicians of differing ability levels can be tested or examined. Other common test formats within the UK and beyond include the BMO (which is a multiple-choice test competition paper used in order to determine the best candidates that are to represent countries within the International Mathematical Olympiad).
↑ GER Lloyd (2009), "What was mathematics in the ancient world? Greek and Chinese perspectives", The Oxford Handbook of the History of Mathematics, Oxford: Oxford University Press, p.12, ISBN9780199213122
In mathematics, a conjecture is a conclusion or a proposition which is suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found. Some conjectures, such as the Riemann hypothesis or Fermat's Last Theorem, have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.
Gregory John Chaitin is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem. He is considered to be one of the founders of what is today known as algorithmic complexity together with Andrei Kolmogorov and Ray Solomonoff. Along with the works of e.g. Solomonoff, Kolmogorov, Martin-Löf, and Leonid Levin, algorithmic information theory became a foundational part of theoretical computer science, information theory, and mathematical logic. It is a common subject in several computer science curricula. Besides computer scientists, Chaitin's work draws attention of many philosophers and mathematicians to fundamental problems in mathematical creativity and digital philosophy.
Mathematics includes the study of such topics as numbers, formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes. There is no general consensus about its exact scope or epistemological status.
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
Mathematical logic is the study of logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
In the philosophy of mathematics, constructivism asserts that it is necessary to find a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation.
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.
In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives. The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts.
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.
Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system that describes a set of sentences that is closed under logical implication. A formal proof is a complete rendition of a mathematical proof within a formal system.
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.
Rigour or rigor describes a condition of stiffness or strictness. Rigour frequently refers to a process of adhering absolutely to certain constraints, or the practice of maintaining strict consistency with certain predefined parameters. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law.
Sixth Term Examination Papers in Mathematics, often referred to as STEP, are university admissions tests for undergraduate Mathematics courses developed by the University of Cambridge.
The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics at the University of Cambridge. It is the oldest Tripos examined at the University.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
Additional Mathematics is a qualification in mathematics, commonly taken by students in high-school. It is applied to a range of problems set out in a different format and wider content to the standard Mathematics at the same level.
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