Is this statement true about mathematical propositions?

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In summary, an axiom is a statement that is assumed but cannot be proven, while a theorem is a statement that can be proven.
  • #1
TruthIsBeauty
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Homework Statement



"The propositions of mathematics work like this: we believe them first, and only later do we try to prove them."

Homework Equations



N/A

The Attempt at a Solution



I read it in some philosophy book but I'm not sure if it's true because I thought the propositions were not believed nor disbelieved until they were proven.

I apologize if I posted in the wrong section. I've just started using forums and I couldn't find any other section that seemed relevant to my question.
 
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  • #2
I don't believe for a moment that is true. Of course, we have to decide which propositions we are going to try to prove and perhaps the author is asserting that those are the ones we "believe" are true. I would say, rather, that they are the ones we suspect are true.
 
  • #3
No, this is not true at all. This is not how mathematics works.
 
  • #4
micromass said:
No, this is not true at all. This is not how mathematics works.

Would you mind explaining why it's not true? I would really appreciate it.
 
  • #5
In mathematics, we don't start off with a large number of propositions that we assume are all true ("believe" to be true), and then later work our way through them, proving that they are true.

How it really works is that we start with a small number of axioms or postulates (e.g., Axiom of Choice, Parallel Postulate - parallel lines never intersect), and using them, derive and prove many more statements (often called theorems). What the proofs do is show that the theorems must be true as long as the initial axiomatic hypotheses are true.

Sometimes mathematicians challenge the axioms, and end up developing new areas of mathematics. For example, non-Euclidean geometry arises from assuming that parallel lines can intersect (see http://en.wikipedia.org/wiki/Non-Euclidean_geometry).

The writer whom you quote lumps all these into one amorphous word: proposition. It's no surprise to me that the writer is not a mathematician.
 
  • #6
Mark44 said:
In mathematics, we don't start off with a large number of propositions that we assume are all true ("believe" to be true), and then later work our way through them, proving that they are true.

How it really works is that we start with a small number of axioms or postulates (e.g., Axiom of Choice, Parallel Postulate - parallel lines never intersect), and using them, derive and prove many more statements (often called theorems). What the proofs do is show that the theorems must be true as long as the initial axiomatic hypotheses are true.

Sometimes mathematicians challenge the axioms, and end up developing new areas of mathematics. For example, non-Euclidean geometry arises from assuming that parallel lines can intersect (see http://en.wikipedia.org/wiki/Non-Euclidean_geometry).

The writer whom you quote lumps all these into one amorphous word: proposition. It's no surprise to me that the writer is not a mathematician.


Thank you very much. I knew something was wrong when the author tried to substantiate his weak argument by using that quote.
 
  • #7
You are correct, the author is wrong
If you want a good book on propositional logic try How to Prove it - A Structured Approach

A proposition is simply a statement that you want to try and prove
The closest thing to what your author described would be an axiom, axioms are assumed but you can't prove axioms since you'd be using the axioms to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using to prove the axioms that you're using
 

1. What are mathematical propositions?

Mathematical propositions are statements that can be either true or false, and can be proven through logical reasoning and mathematical operations. They are often represented using symbols and equations.

2. How are mathematical propositions different from regular statements?

Unlike regular statements, mathematical propositions must be able to be proven true or false using logic and mathematical operations. They are based on objective, verifiable facts and not opinions or beliefs.

3. Can mathematical propositions be proven to be true or false?

Yes, mathematical propositions can be proven to be true or false through logical reasoning and mathematical operations. However, some complex propositions may require advanced mathematical techniques to prove.

4. Are all mathematical propositions important?

Not all mathematical propositions are important, as some may be too simple or obvious to be of significance. However, many mathematical propositions have practical applications in various fields such as science, engineering, and economics.

5. How do we know if a mathematical proposition is true?

A mathematical proposition is considered true if it can be proven through logical reasoning and mathematical operations. Additionally, it must be consistent with other established mathematical theories and concepts.

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