Reasoning and proof in mathematics refer to the process of using logic and evidence to determine the veracity of a mathematical statement or theorem. It involves examining and analyzing the available evidence, using logical deduction to derive conclusions, and providing a clear and rigorous justification for the solution.
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Reasoning and proof in mathematics are fundamental components of mathematical thinking and problem-solving. It involves the critical process of examining and analyzing mathematical evidence, using logical deduction to derive valid conclusions, and providing a clear and rigorous justification for the solution. As mathematician Richard Feynman once said, “You cannot prove a vague theorem because it is simply not true.”
The process of mathematical reasoning and proof is crucial for constructing new mathematical knowledge and for ensuring that mathematical statements and theorems are valid. As stated by the National Council of Teachers of Mathematics (NCTM), “Reasoning and proving are essential to deepen understanding of mathematical concepts and relationships, overcome misconceptions, and make connections among ideas.”
Here are some interesting facts about reasoning and proof in mathematics:
- Euclid’s Elements, written in 300 BCE, is one of the oldest surviving text on mathematics and is famous for its approach to logic and reasoning.
- Proofs can take on various forms, including direct proof, proof by contradiction, proof by induction, and proof by exhaustion. Each type of proof has its strengths and weaknesses depending on the problem at hand.
- The concept of proof was first introduced in ancient Greece by mathematicians such as Euclid and Pythagoras. The concept evolved over time and became a fundamental aspect of modern mathematics.
- Mathematicians and philosophers have debated the nature of proof and what counts as a valid proof for centuries. This has led to advancements in logic and the development of new proof techniques.
- Mathematics competitions such as the International Mathematical Olympiad and the Putnam Competition often require students to provide rigorous proofs for challenging problems. These competitions test not only mathematical ability but also the contestants’ reasoning and proof skills.
In summary, reasoning and proof in mathematics involve critical thinking, logical deduction and rigorous justification. It is a fundamental part of mathematical problem-solving and contributes to the development of new knowledge in the field. As stated by mathematician Emmy Noether, “In mathematics, the art of asking questions is more valuable than solving problems.”
See a video about the subject.
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.
Other viewpoints exist
According to mathematical reasoning, if we encounter an if-then statement i.e. ‘if a then b’, then by proving that a is true, b can be proved to be true or if we prove that b is false, then a is also false.
Reasoning and Proof
- Recognize reasoning and proof as fundamental aspects of mathematics
- Make and investigate mathematical conjectures
- Develop and evaluate mathematical arguments and proofs
Mathematical reasoning represents an extension of logical reasoning, in that the inference rules from propositional logic are still valid and used in a mathematical proof, but mathematical proofs require the following additional knowledge. De nitions Mathematical statements that require proof usually involve the use of terms that must
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What is reasoning and proofs?
Response to this: Reasoning, without logic, is an innate non-verbal action to make inferences about consequences. Proof is an explanation accepted by a community at a given time. The rigor of the proof is dependent on the members of the community.
What is reasoning in mathematics?
In mathematics, reasoning involves drawing logical conclusions based on evidence or stated assumptions. Sense making may be considered as developing understanding of a situation, context, or concept by connecting it with existing knowledge or previous experience.
What is an example of reasoning in math?
A simple example of inductive reasoning in mathematics.
You could use a multitude of examples. Or you could make the generalization that an odd number is just an even number plus 1. Thus, adding two odd numbers is really just adding two even numbers plus 2 and the sum of two even numbers is always even.
What is the definition of proof in math?
As a response to this: A rigorous mathematical argument which unequivocally demonstrates the truth of a given proposition. A mathematical statement that has been proven is called a theorem.
What is mathematical reasoning writing and proof?
The response is: Mathematical Reasoning: Writing and Proof is designed to be a text for the first course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics.
Is statistical proof a mathematical proof?
Answer: While using mathematical proof to establish theorems in statistics, it is usually not a mathematical proof in that the assumptions from which probability statements are derived require empirical evidence from outside mathematics to verify.
What is math reasoning?
As an answer to this: Let’s start with the definition of math reasoning. Reasoning in math is the process of applying logical and critical thinking to a mathematical problem in order to make connections to work out the correct strategy to use (and as importantly, not to use) in reaching a solution.
What is a proof based on?
Response to this: A proof must use correct, logical reasoning and be based on previously established results. These previous results can be axioms, definitions, or previously proven theorems. These terms are discussed in the sections below.