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mathematical proof

n. proof of a mathematical theorem

Mathematical proof

In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture.

Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

Usage examples of "mathematical proof".

By utilizing substantial contributions of the mathematicians Maxim Kontsevich, Yuri Manin, Gang Tian, Jun Li, and Alexander Givental, Yau and his collaborators Bong Lian and Kefeng Liu have finally found a rigorous mathematical proof of the formulas used to count spheres inside Calabi-Yau spaces, thereby solving problems that have puzzled mathematicians for hundreds of years.

The fact that I was able to work out the mathematical proof that you referred to shows only that the letter was written in a language that does not belong to the category of the language we are now using.

Successive frames in this simulation simply correspond to increments in logical extension-like steps in a mathematical proof, adding successive layers of consequences to an initial set of premises.

He loved the strict rigor of a mathematical proof-a string of equations, statements of truth, which nevertheless, if manipulated correctly, led to a deeper, richer truth.

First, under proper tutelage, you must learn how to read the mathematical proof.

Athor and Beenay, you have mathematical proof that Darkness is coming soon.

The Ethical Equations, of course, link conduct with probability, and give mathematical proof that certain patterns of conduct increase the probability of certain kinds of coincidences.

But he went on anyway, for he was never a man to leave a mystery partially solved, a mathematical proof only half worked-out.

There are those who complain that this process does not constitute a mathematical proof, as that term is usually understood, but rather falls more into the category of an experiment, understandably something of a novelty in the field of abstract mathematics.