staymathematicsThe proof is based on a specificAxiomatic systemAccording to certain rules or standardsaxiomandtheoremDerive somepropositionProcess.Compared with evidence, mathematical proof generally depends onDeductive reasoning, rather than relying onNatural inductionAnd empirical reasons.The proposition thus derived is also called the theorem in the system.
Mathematical proof is based onlogicAbove, but usually containsnatural language, so there may be some ambiguous parts.In fact, if most of the proof is written in mathematical form, it can be regarded as the application of informal logic.In the category of proof theory, only the proofs written in pure formal language are considered.This difference has led to most tests of past and present mathematical practice, mathematical quasi empiricism and folk mathematics (or popular mathematics).Mathematical philosophy focuses on the role of language and logic in mathematical proof, and mathematics as language.
mathematicsThe proof on includes two different concepts.The first is informal proof: a rigorous proof used to persuade the audience or readers to accept a certain theorem or assertionnatural languageexpression.Since this kind of proof depends on the language used by the prover, the rigor of the proof will depend on the language itself and the understanding of the audience or readers of the language.Informal proof appears in most applications, such aspolular scienceLectures, oral debates, parts of primary or higher education.Sometimes informal proofs are called "formal" because they are rigorous and well grounded, butmathematical logicWhen scholars use "formal" proof, they mean another kind of completely different proof - formal proof.
staymathematical logicFormal proof is notnatural languageWriting, but written in a formal language: this language is composed of a fixedalphabetConsisting of a string formed by characters in.The proof is a finite length sequence expressed in formal language.This definition makes the formal proof without any logical ambiguity.The theory of formalization and axiomatization proved by research is calledProof theory。Although theoretically, every informal proof can be transformed into a formal proof, it is rarely used in practice.The research on formal proof is mainly applied to the nature of provability in a broad sense, or to the non provability of some statements.[1]
requirement
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The object of proof is proposition, the essence of proposition is assertion, and the nature of assertion is clear.A clear explanation is that there is no ambiguity.Many mathematical proofs cannot be regarded as complete proofs if the result of fuzzy concepts occurs.Therefore, mathematical proof requires mathematical concepts to be accurate, specific, systematic, stable, verifiable and distinguishable.The reasoning meets the requirements of formal logic.In other disciplines, such as physics, scientific facts can soon rise to scientific laws.However, mathematical proof does not recognize scientific facts (so induction is invalid), and mathematical theorems can only be calculated after deductive proof of scientific concepts in fact..Only through strictLogical proofTo confirm the truth of the conclusion is the most fundamental difference between mathematics and other disciplines.
1. Mathematical proof has direct positive proof, that is, deduction refers to the syllogism method, which has 256 lattices and only 24 effective lattices.
The most commonly used lattice for full name judgment generated by mathematical proof is AAA.
The syllogism method must strictly follow the rules, seesyllogism。,
2. It is most likely to make mistakes when using the method of disproof in mathematical proof.Because absurdity uses "hypothesis".
1. Assumption.It can only be used in the proof of negative results, for example, Euclid proved that there are infinite prime numbers and FermatInfinite descent method。 Assuming that a is true, we can deduce b, and get that c, c and a are contradictory, so the assumed a cannot be true, and we get non a.
2. Assumptions cannot be used in positive conclusions.Assuming a, we can deduce b and get c, c=a, or c contains a, so the assumed a is true.(This is the error of the expected reason)
3. Why can "hypothesis" only be used for negative conclusions, but not for positive ones?A stronger logical constraint to scientific theories is that they can be falsified.In other words, because it can be observed and tested meaningfully in the future, the theory must have the possibility of being falsified.This criterion of falsification is a way to distinguish science from pseudoscience.The reason lies in the inherent limitation of confirmation, which can only increase the credibility of one theory, but cannot prove the complete correctness of the whole theory.Because at some point in the future, there will always be cases that conflict with the theory.Only through strict logical proof can we confirm the truth of the conclusion, which is the most fundamental difference between mathematics and other disciplines.
Object of certification
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The object of proof refers to the content to be proved, that is, the main item.For example, "There are infinite prime numbers".The main item is "prime number".The main item can only be a single concept and a universal concept.A separate concept is a unique concept, such as "Shanghai".Because all mathematical theorems are universal judgments, and the main items of all universal judgments are universal concepts and individual concepts.
Universal concept
It reflects the concept of more than one object, and reflects a "class". The connotation of this term consists of the nature of things that must be included in the extension of the term.
UniversalEvery individual must haveThe basic properties of this concept.
For example, "worker" is a universal concept, whether "oil worker", "steel worker", or“Chinese workers”, "German workers", they inevitably have the basic attribute of "workers".General concepts in mathematics include "prime number", "composite number", etc.
"There are infinitely many prime numbers" is the proposition of a universal concept.
Separate concept
It is a unique concept with only one extension, such as "Shanghai" and "Sun Yat sen".Separate concepts in mathematics include "e" and "π".
"E is a transcendental number" is a proposition of a separate concept.
Set concept
The concept of set reflects the concept of aggregate. The extension of this term is composed of the set of things applied to the term. For example, "the Chinese working class" is a concept of aggregateEvery individual is not necessarilyHaving the basic attribute of aggregation, such as a "Chinese worker", does not necessarily have the basic attribute of "Chinese working class"[12]。In mathematics, the proposition that the main term is the concept of a set includes Fermat's big theorem and Riemann's conjecture.It should also be noted that "set concept" refers to an aggregate, and the individuals in the aggregate do not necessarily have the basic attributes of the aggregate. Therefore, the proof of the concept of set must use complete induction to prove each individual one by one.Strictly speaking, the concept of set is not called "proof", but induction.(See any book of Logic).
A formula is the difference between a set concept or a general concept
(1) , general concept proposition formula
There is no variable in the formula, or there is a variable n and it can be infinite, but the nature of things can be judged according to the calculation results, which is a general concept proposition formula.
The formula of general concept,Know the nature of the calculation results before calculation。For example, we can see that a ²+b ²=c ² is a right triangle.
(2) , set concept formula
The characteristic is that it is impossible to know the nature of a numerical result until a specific numerical value is proved or calculated.
For example, Euler's prime number formula in 1772 is a set concept formula: f (n)=n ²+n+41
The values of are prime numbers.
For the first few natural numbers n=0, 1, 2, 3..., the polynomial value is 41, 43, 47, 53, 61, 71.When n equals 40, the value of the polynomial is 1681=41 × 41, which is a composite number.In fact, when n can be divided by 41, P (n) can also be divided by 41, so it is a composite number.
The formula of set concept cannot guarantee that the calculation result has the desired result property of the formula, and it is an uncertain result formula.Because each individual of the concept of set does not necessarily have the basic attributes of the concept.
II. Classification by attributes or entities
1. Attribute concept (e.g. prime number; irrational number).
2. Entity concept (such as a formal structure, binomial).
3. Attribute contains entity (Fermat prime number).
4. Entity contains attribute (twin prime number).
III. According to the logical hierarchy
First order logic (all mathematical theorems are first order logic)
Second order logic (refers to the rate of change, such as Riemann conjecture and Fermat's big theorem)
standard
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Mathematical proof must be strictly in accordance with unified standards
1. The object of proof must be a universal concept, and it is not allowed to "prove" the set concept.
2. The proof method must be correct deductive proof（Mathematical inductionThe mathematical induction method without unified formula is invalid under the formula that can unify all element objects of this universal concept).
3. The argument must be correct.
4. Do not use fuzzy concepts, that is, concepts must be the only interpretation, and there must be no ambiguity (for example, the so-called "almost prime number", "sufficiently large", etc. are strictly prohibited).
5. All conclusions must be operable, that is to say, after the conclusion is proved, people can know the result through the calculation of this conclusion, without contradictory results.
6. The conclusion must be in full name, and the special conclusion is invalid.
7. The proof process must be transitive, and the proof without transitivity is invalid. For example, in the process of proving Fermat's big theorem, the Fermat's big theorem and the Tanyama Zhicun conjecture are not transitive, so the proof is invalid.
The transitive relationship is a special relationship, referring to A and B;B and C;,Both, we can infer that A and C also have.
As for the transmission relationship, A and B are brothers, and B and C are brothers. Therefore, A and C are brothers.
Anti transitive relationLao Zhang is the father of Da Zhang and Da Zhang is the father of Xiao Zhang, so Lao Zhang is not Xiao Zhang's father (father should also be strictly defined, see the above).
The non transitive relationship is mistaken for the anti transitive relationship: 100 meters from ground a to ground b and 100 meters from ground b to ground c, so it will not be 100 meters from ground a to ground c.(How far apart is a non transitive relationship, which is mistaken for an anti transitive relationship. For example, three vertices of an equilateral triangle are equal)
Logical relationship between concepts
Greater than relation
Antisymmetric relation
Transitive relationship
Less than
Antisymmetric relation
Transitive relationship
Identical relationship between concepts
Symmetry relation
Transitive relationship
Inclusion between concepts
Asymmetric relation
Transitive relationship
Cross relations between concepts
Symmetry relation
Nontransitive relation
Completely different relationship between concepts
Symmetry relation
Nontransitive relation
Contradictory relationship
Symmetry relation
Anti transitive relation
Opposition relationship
Symmetry relation
Nontransitive relation
Implication relation
Asymmetric relation
Transitive relationship
Inverse implication relation
Asymmetric relation
Transitive relationship
Equivalence relation
Symmetry relation
Transitive relationship
Proof required
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The first level of mathematics is mathematical facts, for example, 3 and 5 are prime numbers;
The second level of mathematics is mathematical concept, which is to summarize facts into a systematic meaning.For example, "twin prime number";Refers to two prime numbers with a difference of 2.
The third level of mathematics is mathematical theorem, which needs to be proved from mathematical concept to mathematical theorem.For example, it is a theorem that there are infinitely many Euclidean prime numbers.For example, the proposition "There are infinitely many prime twins" has not been proved yet.
The fourth level of mathematics is mathematical theory.For example, [Elementary Number Theory] includes a series of concepts, theorems, formulas, images and functions.
The first level will not automatically rise to the third level. It must be completed by deductive proof with the help of the second level, namely concept.
Mathematical status
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The whole content of mathematics is divided into three categories. The first is mathematical imagination, which is used to expand the scope of mathematical research.The second type is mathematical calculation.The third type is mathematical proof.Imagination must conform to things and internal logic.The calculation must be as correct as possible.Proof needs to be rigorous and logical.The three categories intersect each other.The mathematical imagination is high (see farther), the mathematical calculation is rich (master rich calculation methods), and the mathematical proof is handsome (the commander in chief is handsome, without him, the calculation and imagination are unreliable).
