## View Poll Results: Is the deduction from an inconsistent set of premises in the argument in the OP, valid?

Voters
3. You may not vote on this poll
• Yes, it is valid.

2 66.67%
• No, it is not valid.

1 33.33%

0 0%
Multiple Choice Poll.

# Thread: The square root of two (an argument with inconsistent premises)

1. To keep up my nitpicking, this isn't algebra (which to a modern mathematician, means abstract algebra). This is basic number theory, or arithmetic, in which one of the most important early theorems is the uniqueness of prime factorization, also known as "The Fundamental Theorem of Arithmetic".  Reply With Quote

2. Originally Posted by A Toy Windmill To keep up my nitpicking, this isn't algebra (which to a modern mathematician, means abstract algebra). This is basic number theory, or arithmetic, in which one of the most important early theorems is the uniqueness of prime factorization, also known as "The Fundamental Theorem of Arithmetic".
I studied the Fundamental Theorem of Arithmetic (and related matters) in a course named "Algebra 1", and that is standard over here. It's essentially one of the first things one studies in college. The sort of fine-grained distinction you are making is not generally made a such an elementary level. Courses focusing only on number theory are much more advanced. But I guess it's probably different over there, so he'd likely have to take a course in number theory to understand the proof? That's even less likely to happen. Oh well.   Reply With Quote

3. I think I probably first came across the proof in a course called something like "introducing mathematics". It's still number theory --- being a theorem in the theory of natural numbers --- even if very basic. And most of that textbook you linked is number theory too.  Reply With Quote

4. Originally Posted by A Toy Windmill I don't think you understand the turnstile relation.
I don't think you understand logic at all.
EB  Reply With Quote

5. Originally Posted by Angra Mainyu  Originally Posted by Speakpigeon I voted "not valid".
It's a standard proof. So, you wanted to know about the impact of having what you believe to be the "wrong" logic on mathematics? Well, as I pointed out, it's pervasive. This proof is not weird, or suspect, or anything like that. This is (pretty much, not every detail; I don't remember every detail, plus I simplified it a little because the only prime I consider is 2) a proof that was given in a basic algebra course I took (when talking about the applications of prime factorization, iirc). That was before I took a course in mathematical logic. I understood the proof. All of the other students understood the proof. Very probably (almost certainly), none of them had taken a course in mathematical logic, either (that's generally for considerably more advanced students). We all reckoned the proof was correct. That is our intuitive sense of logic.

If you want evidence that this is a standard proof, then look it up: look for proofs of the irrationality of the square root of two, and you'll see that, other than details, they're essentially of the same sort. I don't think you will find any that is valid in your logic. For example: just google "square root 2 irrational" without quotation marks to see the arguments.
I don't have any problem with the mathematical proof.

And I'm sure your interpretation of it in terms of mathematical logic is just wrong.

So, no, it's not an example of the wrong logic of mathematical logic applied to a mathematical theorem.
EB  Reply With Quote

6. Originally Posted by A Toy Windmill I think I probably first came across the proof in a course called something like "introducing mathematics". It's still number theory --- being a theorem in the theory of natural numbers --- even if very basic. And most of that textbook you linked is number theory too.
I'm not saying it's not number theory. I linked to the textbook (titled "Algebra 1") because those are the notes of the course by the same name (which I also linked to), and its content is to a considerable extent the content of the introductory course I took. There are some differences because I took the course long ago and it was updated later, but the author of the notes was my professor on that course, and the content - including the part about the Fundamental Theorem of Arithmetic - were part of the course back then, even if with a somewhat different approach on some matters.

There is a much more advanced course called "Teoría de Números" ("Number Theory"), but that course (and any others involving number theory) is also part of a broader category titled 'algebra'. I see the word is used much more restrictively over there. Fair enough, so steve_bank would get that proof in a basic number theory course, not a basic algebra course.   Reply With Quote

7. Originally Posted by Speakpigeon  Originally Posted by Angra Mainyu  Originally Posted by Speakpigeon I voted "not valid".
It's a standard proof. So, you wanted to know about the impact of having what you believe to be the "wrong" logic on mathematics? Well, as I pointed out, it's pervasive. This proof is not weird, or suspect, or anything like that. This is (pretty much, not every detail; I don't remember every detail, plus I simplified it a little because the only prime I consider is 2) a proof that was given in a basic algebra course I took (when talking about the applications of prime factorization, iirc). That was before I took a course in mathematical logic. I understood the proof. All of the other students understood the proof. Very probably (almost certainly), none of them had taken a course in mathematical logic, either (that's generally for considerably more advanced students). We all reckoned the proof was correct. That is our intuitive sense of logic.

If you want evidence that this is a standard proof, then look it up: look for proofs of the irrationality of the square root of two, and you'll see that, other than details, they're essentially of the same sort. I don't think you will find any that is valid in your logic. For example: just google "square root 2 irrational" without quotation marks to see the arguments.
I don't have any problem with the mathematical proof.

And I'm sure your interpretation of it in terms of mathematical logic is just wrong.

So, no, it's not an example of the wrong logic of mathematical logic applied to a mathematical theorem.
EB
You said the proof is not valid. Now you're saying it is valid? Is it valid, or invalid? If it's valid, why did you vote "invalid"? If it's invalid, how come you have no problem with the mathematical proof?  Reply With Quote

8. Originally Posted by Speakpigeon
And I'm sure your interpretation of it in terms of mathematical logic is just wrong.
What "interpretation in terms of mathematical logic" are you even talking about?
I gave a proof. It is correct. It is a standard proof. You claimed it is not valid, and voted so. Later you said "I don't have any problem with the mathematical proof." What do you even mean?  Reply With Quote

9. Originally Posted by Angra Mainyu  Originally Posted by steve_bank 2^1/2 = n/m
N cannot be less than or equal m otherwise you get a number less than 1 or equal to 1.
N and m must both be positive or negative.
For n > m n cannot be an integr number of m.

And so on. The problem is not properly bounded.

2^1/2 = n/m
N^2 = 2m^2
n = sqrt(2m^2)
n/m = 2 ^1/2

The manipulation says nothing about whether it is solvable. That is not a proof. You transpositions of m and n are not valid, the original form remains.

The question is can any arbitrary irrational number be expressed as the ratio of two integers. I think there was a math thread on this.

This implies that n12 is even. Hence, n1 is even. Therefore, there is an integer n2 such that 2m1^2 = n1^2
No it does not. You have to show that if this were a mathematical proof.
I regret you do not understand it, but it is a standard proof. It's the sort of proof you'll get in a basic algebra course, when talking about prime factorization. I do not understand why you are not able to follow it (I simplified it, so that only division by 2 is considered, leaving aside other primes), but I suggest you just google "square root two not rational" (without the quotation marks) or something like that, and you will find proofs pretty much like this one, except for details.

You proved nothing. You sarted with something from a book and tried to turn into a logical question. Was the goal to prove the existence of an integer ratio that equals sqrt(2)?

When you said it follows that.., exactly why does it follow. Telling me to Google it says you really do not understand.

Your presentation is messy and imprecise. You made a simple algebraic operation, squaring both sides, and drew a conclusion of something based on a substitution of variables.

If the OP is to demonstrate problems with syllogistic logic, that is not new. The simple premise conclusions syllogism has many problems, which is why modem mathematical logic evolved.

Does sqrt(2) have an integer ratio? If so what is it?  Reply With Quote

10. Originally Posted by steve_bank  Originally Posted by Angra Mainyu I regret you do not understand it, but it is a standard proof. It's the sort of proof you'll get in a basic algebra course, when talking about prime factorization. I do not understand why you are not able to follow it (I simplified it, so that only division by 2 is considered, leaving aside other primes), but I suggest you just google "square root two not rational" (without the quotation marks) or something like that, and you will find proofs pretty much like this one, except for details.

You proved nothing. You sarted with something from a book and tried to turn into a logical question. Was the goal to prove the existence of an integer ratio that equals sqrt(2)?

When you said it follows that.., exactly why does it follow. Telling me to Google it says you really do not understand.

Your presentation is messy and imprecise. You made a simple algebraic operation, squaring both sides, and drew a conclusion of something based on a substitution of variables.

If the OP is to demonstrate problems with syllogistic logic, that is not new. The simple premise conclusions syllogism has many problems, which is why modem mathematical logic evolved.

Does sqrt(2) have an integer ratio? If so what is it?
Have a good day.  Reply With Quote

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