# Question — why is mathematical proof hard?

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Mathematical proof is hard because it involves rigorous logical reasoning and the need to establish the truth of a statement beyond any doubt, which may require advanced techniques and significant creativity.

## Response to your request in detail

Mathematical proof is hard due to the rigorous logical reasoning required to establish the truth of a statement beyond any doubt. It is not enough to show that a statement is likely true; rather, a mathematical proof must provide a logical argument that demonstrates the truth beyond any doubt. This often requires advanced techniques and significant creativity.

As Richard Feynman, a renowned physicist and mathematician, famously stated, “To mathematics, the art of handling symbols with reason, has come the puzzle of making the reasoning itself foolproof.”

Interesting facts on mathematical proof include:

• The concept of mathematical proof dates back to ancient Greece, where mathematicians such as Euclid and Pythagoras used logical reasoning to prove theorems.
• In addition to being used in mathematics, proof is used in other fields including science, engineering, and law.
• There are many different types of mathematical proofs, including direct proof, proof by induction, proof by contradiction, and constructive proof.
• The process of writing a proof can often involve trial and error, as well as multiple revisions and adjustments.
• While some mathematical proofs are relatively straightforward, others can take years or even centuries to develop.

Table: Common Types of Mathematical Proof

Type of Proof Explanation
Direct Proof A proof that establishes that a statement is true by directly applying the definition or axioms of a mathematical system.
Proof by Contradiction A proof that establishes that a statement is true by proving that assuming the opposite of the statement leads to a contradiction.
Proof by Induction A proof that establishes that a statement is true for all values in a sequence of numbers by proving that the statement is true for a base case and for any other case assuming the statement is true for the previous case.
Constructive Proof A proof that not only establishes that a statement is true, but also provides an example or algorithm for constructing the object that the statement describes.

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## I discovered more answers on the internet

12 Answers. You can’t learn "how to prove". "Proving" is not a mechanical process, but rather a creative one where you have to invent a new technique to solve a given problem. A professional mathematician could spend their entire life attempting to prove a given statement and never succeed.

Mathematical proofs are difficult to construct and require a basic foundation in the subject to come up with the proper theorems and definitions to logically devise a proof. Proving is not a mechanical process, but rather a creative one where you have to invent a new technique to solve a given problem. A professional mathematician could spend their entire life attempting to prove a given statement and never succeed. Proofs are inherently hard.

Unfortunately, there is no quick and easy way to learn how to construct a proof. You must have a basic foundation in the subject to come up with the proper theorems and definitions to logically devise your proof. By reading example proofs and practicing on your own, you will be able to cultivate the skill of writing a mathematical proof.

You can’t learn "how to prove". "Proving" is not a mechanical process, but rather a creative one where you have to invent a new technique to solve a given problem. A professional mathematician could spend their entire life attempting to prove a given statement and never succeed.

So, yes, proofs are actually inherently hard! Things are even worse; if you believe that S does not prove any false arithmetic sentence (which you should otherwise why are you using S?), then we can explicitly construct an arithmetical sentence Q such that S proves Q but you must believe that no proof of Q over S has less than 210000 symbols!

I feel like i am memorizing the proofs rather than learn how to prove

You can’t learn “how to prove”. “Proving” is not a mechanical process, but rather a creative one where you have to invent a new technique to solve a given problem. A professional mathematician could spend their entire life attempting to prove a given statement and never succeed.

I can easily deal with any type of proofs that i saw before ( eg. once i saw the proof of a recurrence question i became pretty good at prooving them). My problems start when i face an unusual question.

That is normal. Any mathematics “proofs” course isn’t designed to teach you how to take an arbitrary problem you’ve never seen before and be able to solve it (since nobody, not even the best mathematics professors can do that). Rather, your learning goals are
• Learn how to “read” proofs and judge their correctness
• Learn how to “write” down a proof in the right mathematical language
• Learn about known proof “techniques” and how to apply …

These topics will undoubtedly pique your attention

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Subsequently, Are proofs in math hard?
The reply will be: 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.

What is the point of a mathematical proof?
As an answer to this: According to Bleiler-Baxter & Pair , 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.

How to do well in math proofs?
As a response to this: There are 3 main steps I usually use whenever I start a proof, especially for ones that I have no idea what to do at first:

1. Always look at examples of the claim. Often it helps to see what’s going on.
2. Keep the theorems that you’ve learned for an assignment on hand.

What is the world’s longest math proof?
The proof, which concerns the classification of mathematical symmetry groups – a concept aptly known as the "Enormous Theorem" – took 100 mathematicians three decades and some 15,000 pages of workings to pin down.

Besides, Why are a substantial number of mathematical proofs hard? The answer is: A substantial number of mathematical proofs are hard because they require new ways of thinking. I like to say that a large number of mathematical proofs require you to figure out a way to do something that would take an infinite amount of time in a finite amount of time, and that’s where the difficulty lies.

Also question is, What is a mathematics proofs course? Any mathematics "proofs" course isn’t designed to teach you how to take an arbitrary problem you’ve never seen before and be able to solve it (since nobody, not even the best mathematics professors can do that). Rather, your learning goals are Learn how to "write" down a proof in the right mathematical language

What is the most difficult case for proofs?
The reply will be: For proofs you don’t know whether there is such a proof from your start is even harder and not even knowing where to start is the most difficult case. There are easy proofs. Like there are people climbing on hills. They don’t put people who climb on hills on the news. They only talk about people who climb the Everest because it’s difficult.

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Is mathematics useful to a variety of people without knowing the proofs?
Response to this: 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".

Why are a substantial number of mathematical proofs hard?
The response is: A substantial number of mathematical proofs are hard because they require new ways of thinking. I like to say that a large number of mathematical proofs require you to figure out a way to do something that would take an infinite amount of time in a finite amount of time, and that’s where the difficulty lies.

Then, Do mathematical proofs need to be rigorous?
On the one hand, mathematical proofs need to be rigorous. Whether submitting a proof to amath contest or submitting research to a journal or science competition, we naturally want it to becorrect. One way to ensure our proofs are correct is to have them checked by a computer.

Besides, What is a proof in maths?
The response is: Most students starting out in formal maths understand that a proof convinces someone that something is true, but they use the same reasoning that convinces them that everyday things are true: empirical reasoning.

Likewise, What is the difficulty of proof? Answer: “The difficulty… is to manage to think in a completely astonished and disconcerted way about things you thought you had always understood.” ― Pierre Bourdieu, Language and Symbolic Power, p. 207 Proof is the central epistemological method of pure mathematics, and the practice most unique to it among the disciplines.

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