# What do you inquire: are formal mathematical proofs deductive in nature?

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Yes, formal mathematical proofs are deductive in nature, as they use a series of logical steps to arrive at a conclusion from a set of premises or axioms.

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Yes, formal mathematical proofs are deductive in nature, as they use a series of logical steps to arrive at a conclusion from a set of premises or axioms. This deductive approach ensures that the conclusion reached is logically sound and can be applied universally.

One of the most famous mathematicians, Euclid, used deductive reasoning to create his famous work, “Elements.” According to Euclid, only through logical proofs and deductions could conclusions be accurately made about the nature of mathematics.

Furthermore, formal mathematical proofs often involve the use of symbols and notation to represent logical operations and mathematical concepts. These symbols and notations enable mathematicians to convey complex ideas and reasoning in a concise and precise manner.

In addition, mathematical proofs are often subject to peer review by other mathematicians, who ensure that the proofs are logically consistent and accurate. This process helps to eliminate errors and inaccuracies in mathematical research and ensures that the conclusions reached are reliable and valid.

Overall, formal mathematical proofs are an essential tool for mathematicians and anyone involved in mathematics to ensure that conclusions are reached logically and with the highest degree of accuracy and precision possible.

As Albert Einstein once said, “Pure mathematics is, in its way, the poetry of logical ideas.” Through the use of deductive reasoning and logical proofs, mathematicians are able to explore the beautiful and intricate nature of mathematics and the universe as a whole.

| Benefits of Formal Mathematical Proofs |
| — | — |
| Ensures logical consistency |
| Eliminates errors and inaccuracies |
| Enables concise and precise expression of ideas |
| Enables universal application of conclusions |
| Subject to peer review for accuracy |

Table: The Benefits of Formal Mathematical Proofs

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The video provides an introduction to inductive and deductive reasoning. The concept of inductive reasoning is explained using a basket of mangoes as an example, where a general conclusion is drawn based on specific observations. However, the video notes that although a conclusion may be logically true, it may not necessarily be realistically true. To illustrate this point, a second example about a box containing fruits is given, where the conclusion drawn from two true statements is wrong. It is also noted that inductive reasoning is frequently used in mathematics to arrive at conjectures that need to be proved with specific cases, using the principle of mathematical induction.

## View the further responses I located

A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.

At its core, the question you are asking (when properly formulated) is interesting, difficult and poorly understood.

The first issue, as discussed in the comments, is that one has to differentiate between different stages:
• How do mathematicians come up with conjectures or guesses what’s true? (Before something is proven, it is not called a theorem but a conjecture.) What plays the role of “nature” or “experimental evidence” in mathematics, when compared to other sciences?
• How do mathematicians come up with proofs of their (or somebody else’s!) conjectures?
• What’s the nature of a formal mathematical proof?
• How do mathematicians explain their proofs to others or/and convince others that their proofs are correct?

Only stage 3 is deductive. See, for instance, this question and answers.

There are no definitive answers for 1, 2 and 4. Poincare was very interested in 1 and 2 and discussed these (based on his own experience)
in his “Reflections on Mathematical Creation”. One can say…

## More intriguing questions on the topic

What is an example of a deductive proof?
The reply will be: With this type of reasoning, if the premises are true, then the conclusion must be true. Logically Sound Deductive Reasoning Examples: All dogs have ears; golden retrievers are dogs, therefore they have ears. All racing cars must go over 80MPH; the Dodge Charger is a racing car, therefore it can go over 80MPH.

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Are mathematical proofs deductive? Answer will be: A deductive argument is characterized by the claim that its conclusion follows with strict necessity from the premises. A mathematics proof is a deductive argument. Although induction and deduction are processes that proceed in mutually opposite directions, they are closely related.

Also, Are math proofs inductive or deductive?
Answer: Mathematics is deductive. To be more precise, only deductive proofs are accepted in mathematics. Your "inductive proof" of the distributive property wouldn’t be accepted as a proof at all, merely as verification for a finite number of cases (1 case in your question).

Regarding this, What is the concept of a formal proof?
As a response to this: In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference.
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Beside this, What is natural deduction in logic and proof theory?
Response: In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning .

How does a formal proof differ from a natural language argument?
Response: It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system.

Beside above, How are proofs written?
Response will be: Proofs employ logic expressed in mathematical symbols, along with natural language which 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.

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Is math a deductive science? Response will be: Yes it is a deductive science and No, new "mathematical facts" are conjectured/intuited before than proved. At its core, the question you are asking (when properly formulated) is interesting, difficult and poorly understood. The first issue, as discussed in the comments, is that one has to differentiate between different stages:

Herein, What is natural deduction in logic and proof theory? In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning .

How does a formal proof differ from a natural language argument?
It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system.

How are proofs written?
Proofs employ logic expressed in mathematical symbols, along with natural language which 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.

Likewise, What is the best natural deduction formalism? As a response to this: The best sort of natural deduction formalism is by means of a set of Int-Elim rules, ones that will lead us to normalizability of proofs and will show a harmony in their formalizations. Given the way that such Int-Elim rules operate in formal derivations, they should be organized into harmonious pairs.

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