Argument
Mathematical proof
Overview
A mathematical proof is a deductive 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 that establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning that 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. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
Proofs employ logic expressed in mathematical symbols, along with natural language that usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement 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, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Hence, in the conclusion of a syllogism we restrict a predicate to a certain object, after having thought it in the major in its whole extent under a certain condition. This complete quantity of the extent in relation to such a condition is called universality (universalitas). To this corresponds totality (universitas) of conditions in the synthesis of intuitions.
Immanuel Kant, The Critique of Pure Reason
States what Mathematical proof is; the author is tied to Mathematical proof in the knowledge graph.
The exercise of logical reason is always concerned with these absolutely general conditions. In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key to it. This pattern of…
Alfred North Whitehead, Science and the modern world
States what Mathematical proof is.
In Logic we ordinarily use the expressions demonstrable or indemonstrable only in respect of _propositions_, but these might be better designated by the titles respectively of _mediately and immediately certain_ propositions; for pure Philosophy has also propositions of both kinds, _i.e._ true propositions, some of which are susceptible of proof and others not. It can, as philosophy, prove them on _a priori_…
Immanuel Kant, Kant's Critique of Judgement
States what Mathematical proof is; the author is tied to Mathematical proof in the knowledge graph.
Philosophers who discussed Mathematical proof
Related ideas
- Formal language
- Four color theorem
- Proof assistant
- Argument
- Informal logic
- Infinite set
- Counterexample
- Integer
- Immanuel Kant
- Ancient Greek mathematics
- Latin
- Proof theory
- Mathematical induction
- Geometry
- Mathematical beauty
- Mathematical object
- Mathematical logic
- Rational number
- Natural language
- Natural number
- Number theory
- Reductio ad absurdum
- Probability theory
- Euclid's Elements
Place in the tradition
Part of Aesthetics.