Yes, proof based math is useful as it provides logical and rigorous reasoning for mathematical concepts and theorems, allowing for greater understanding and application in various fields such as science, engineering, and finance.
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Proof based mathematics relies on the use of logical reasoning to support mathematical concepts and statements. While some may argue that memorization of formulas and procedures is enough for application in certain fields, the importance of proof-based mathematics cannot be understated.
As the great mathematician Blaise Pascal once said, “Mathematics is the only science where one never knows what one is talking about nor whether what is said is true.” By providing rigorous, logical and unambiguous proofs for mathematical concepts and theorems, proof-based mathematics allows for a greater understanding and certainty in mathematical knowledge. This translates into applications in various fields such as science, engineering, and finance.
Here are some interesting facts on the topic of proof-based mathematics:
- The ancient Greeks are widely considered to be the pioneers of proof-based mathematics, with notable mathematicians such as Euclid and Pythagoras laying the foundations for modern mathematical proof.
- Proof-based mathematics is also known as axiomatic mathematics, as it is based on the use of axioms (self-evident or accepted statements) to derive further statements and theorems.
- The use of proof-based mathematics has led to the development of various branches of mathematics such as algebra, calculus, and number theory.
- One of the most famous and intricate proofs in mathematics is Andrew Wiles’ proof of Fermat’s Last Theorem, which he spent seven years developing.
- In 2014, Maryam Mirzakhani became the first woman to win the prestigious Fields Medal, often considered the highest honor in mathematics, for her pioneering work in the field of hyperbolic geometry and complex analysis.
To better understand the importance of proof-based mathematics, here is a table comparing traditional mathematical knowledge with knowledge based on proof:
| Traditional Mathematical Knowledge | Proof-Based Mathematical Knowledge |
|---|---|
| Based on memorization of formulas and procedures | Based on logical reasoning and rigorous proofs |
| Focuses on application | Focuses on understanding and certainty |
| May be limited in scope to specific applications | Allows for flexibility in application across fields |
| Does not require a deep understanding of mathematical concepts | Requires a deep understanding of mathematical concepts |
In conclusion, proof-based mathematics is an essential part of mathematical knowledge and provides a foundation for applications in various fields. As Albert Einstein once said, “Pure mathematics is, in its way, the poetry of logical ideas.”
Watch a video on the subject
The video discusses the importance of going back and identifying key results and techniques to solve math proofs. The speaker advises spending time on a problem, but if it becomes too challenging, one should analyze the solution step by step and rewrite it on their own to understand every step. By doing so, individuals can learn more techniques for future problems and understand the significance of each step in the solution.
There are other opinions
An algorithm can often be proven correct by an induction argument, so it’s really useful to understand how those work. The indirect benefit is that taking proof-based math will improve your problem solving skills. What are the benefits of doing your major in maths?
Not only mathematicians, but you yourself can benefit from learning to do proofs. The skills you develop in learning to prove mathematical statements are useful in many other areas of life. You learn logic, which lets you recognize when a supposed "proof" (whether in math or life) is flawed and shouldn’t be believed.
Proofs are important because proofs are just understanding how we know that something is true.
Yes.
The direct benefit of learning to write mathematical proofs is that it will help you reason carefully about correctness, so that you can write code and know how to prove that your code is correct. Many programmers seem to not have this skill. Notoriously, many binary search implementations in the real world contain bugs. An algorithm can often be proven correct by an induction argument, so it’s really useful to understand how those work.
The indirect benefit is that taking proof-based math will improve your problem solving skills.
More intriguing questions on the topic
Herein, How important are proofs in math?
According to Bleiler-Baxter & Pair [22], for a mathematician, a proof serves to convince or justify that a certain statement is true. But it also helps to increase the understanding of the result and the related concepts. That is why a proof also has the role of explanation.
Are proofs the hardest part of math?
Proof writing is often thought of as one of the most difficult aspects of math education to conquer. Proofs require the ability to think abstractly, that is, universally.
Subsequently, Should I memorize math proofs? Answer to this: No. It usually suffices to understand them. However, if you are expected to be able to regurgitate a proof on a test and you don’t trust yourself to be able to do so based solely on the fact that you understood it, then memorizing an outline of the proof can help.
Do mathematicians use proof assistants?
Response will be: To aid mathematicians in organizing their work, researchers have developed formal proof assistants, which are computer languages that can verify mathematical statements. Unlike humans, formal proof assistants do not make mistakes.
People also ask, What is an example of a mathematical proof? A mathematical proof is an argument that deduces the statement that is meant to be proven from other statements that you know for sure are true. For example, if you are given two of the angles in a triangle, you can deduce the value of the third angle from the fact that the angles in all triangles drawn in a plane always add up to 180 degrees.
Beside this, Is math proof good for a kid?
Answer: Learning how to come up with a math proof is good for a kid. It’s a problem-solving activity. Students believe what they are told by their teachers, especially in math class, and with good reason because math teachers rarely teach an untrue theorem, and proofs often are used instead of good explanations.
Regarding this, What are the benefits of learning to prove mathematical statements?
Response to this: The skills you develop in learning to prove mathematical statements are useful in many other areas of life. You learn logic, which lets you recognize when a supposed "proof" (whether in math or life) is flawed and shouldn’t be believed. See our FAQ section on False Proofs: http://mathforum.org/dr.math/faq/faq.false.proof.html.
Is mathematics useful to a variety of people without knowing the proofs? The answer is: I would agree/concede that mathematics is very useful to a variety of people without knowing the proofs. That is, in fact, the "what" of mathematics is already crazily useful…which is why sometimes we’d care about the "why". Sometimes, to have more confidence in the "what".
Subsequently, What is an example of a mathematical proof? In reply to that: A mathematical proof is an argument that deduces the statement that is meant to be proven from other statements that you know for sure are true. For example, if you are given two of the angles in a triangle, you can deduce the value of the third angle from the fact that the angles in all triangles drawn in a plane always add up to 180 degrees.
Is mathematics useful to a variety of people without knowing the proofs?
I would agree/concede that mathematics is very useful to a variety of people without knowing the proofs. That is, in fact, the "what" of mathematics is already crazily useful…which is why sometimes we’d care about the "why". Sometimes, to have more confidence in the "what".
Then, Is math proof good for a kid?
As a response to this: Learning how to come up with a math proof is good for a kid. It’s a problem-solving activity. Students believe what they are told by their teachers, especially in math class, and with good reason because math teachers rarely teach an untrue theorem, and proofs often are used instead of good explanations.
Correspondingly, What is a proof based on?
Answer will be: 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.