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

[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".

 

[Angra Mainyu]

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.

 

[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.

 

[Speakpigeon]

I don't think you understand the turnstile relation.

I don't think you understand logic at all.
EB

 

[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

 

[Angra Mainyu]

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.

 

[Angra Mainyu]

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?

 

[Angra Mainyu]

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?

 

[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?

 

[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.

 

[Speakpigeon]

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?

I'm bemused I voted not valid!

Of course it is valid. Trivially valid. Obviously valid.

Your discussion with ATW also shows you misunderstand this proof for an instance of contradictory premises. I guess ATW explained to you what's the difference between contradictory and inconsistent but I'm not sure you've understood yet.

This kind of proof is called "by contradiction" but this is misleading and you don't understand logic anyway so you can't tell by yourself.

Still, it is clear I mistook this thread for the other one you started, "Yet another logic question", on the validity of P1...Pn, Q1...Qm implies (P and Q) if P1...Pn implies P, and Q1...Qm implies Q. That one is definitely not valid.
EB

 

[Angra Mainyu]

I'm bemused I voted not valid!

Of course it is valid. Trivially valid. Obviously valid.

Fine, then. It's valid.

Your discussion with ATW also shows you misunderstand this proof for an instance of contradictory premises. I guess ATW explained to you what's the difference between contradictory and inconsistent but I'm not sure you've understood yet.

No, you have not understood my discussion with A Toy Windmill. He did not say there were inconsistent but not contradictory premises. He said there was a single false premise - namely, that the square root of two is a rational number -, because he does not call known facts of number theory "premises". However, that does not change the fact that I derived a contradiction from the assumption that that the square root of two is a rational number, plus some theorems of number theory. So, I said, fair enough, if only one of those statements is called "premise", let's call the other ones "statements", so it remains the case that I derived a contradiction from a set of inconsistent statements. Now, A Toy Windmill did not find that natural, either, so in addition to a terminological disagreement, we seem to have some different view about the philosophy of mathematics, but we have no disagreement about what follows from what, what the proof says, etc.

In fact, if you look at it in a first-order model, this is what happened: using a standard axiomatization of the first-order predicate calculus, from the formulas that (in a standard interpretation) assert the axioms that correspond to Peano's postulates (somewhat weakened for the first-order context, but pretty much those ones), and the formula that (also in a standard interpretation) asserts that there is a rational number n/m such that (n/m)²=2, a contradiction follows. So, the (first-order version of) Peano's postulates, together with that formula, form an inconsistent set of formulas - and hence, under the usual interpretation, an inconsistent set of statements -, and that implies a contradiction. If the audience does not already accept the axioms and some relevant theorems of number theory, then one would have to defend them too, which I did in my reply to A Toy Windmill here.

But in any case, we both agree about the logic of the arguments, and at best, our disagreement is about the philosophy of mathematics, if not merely terminological.

You are misconstruing (grossly) what A Toy Windmill say. He did not say anything about distinguishing between inconsistent set of statements and contradictory ones. In fact, that is a matter of terminology too, but it would be standard to use the terms 'inconsistent set of statements' and 'contradictory set of statements' to mean the same, even if the former is probably more common.

Regardless, given that you claim that

ATW explained to you what's the difference between contradictory and inconsistent

I challenge you to provide evidence in support of that claim, by means of posting the relevant links and/or quoting him (you cannot, since you are just misconstruing his words). In fact, he even say

This is all nitpicking, for which I apologize. I'm just throwing it out there in case it turns out to be relevant in your arguments with Speakpigeon.

Well, I guess you are using his nitpicking as a weapon, by means of grossly misrepresenting his words.

This kind of proof is called "by contradiction" but this is misleading and you don't understand logic anyway so you can't tell by yourself.

I do understand logic. You do not.

 

[A Toy Windmill]

Now, A Toy Windmill did not find that natural, either, so in addition to a terminological disagreement, we seem to have some different view about the philosophy of mathematics, but we have no disagreement about what follows from what, what the proof says, etc.

I hope it's not a difference in philosophy of mathematics.

As I said, there are ways to save the idea that you have proven the irrationality of the square root of 2 from inconsistent statements, but in the way suggested, you conceded that it meant stepping outside the formalism, which is to say that what is "really going on" isn't captured formally anymore. I suggest that's a weak point, given that you're arguing with someone with serious disagreements about the nature of logical validity.

As a final remark, and to throw in some real philosophy of mathematics, how does a logicist interpret your proof? They take logical validity to be powerful enough to cover all the mathematics we care about, and can show in their logics that arithmetical facts are just a subset of logical truths. All the ambient mathematical facts are now logical truths, and your proof shows that the statement that the square root of 2 is rational is analytically inconsistent with itself!

 

[Angra Mainyu]

As I said, there are ways to save the idea that you have proven the irrationality of the square root of 2 from inconsistent statements, but in the way suggested, you conceded that it meant stepping outside the formalism, which is to say that what is "really going on" isn't captured formally anymore. I suggest that's a weak point, given that you're arguing with someone with serious disagreements about the nature of logical validity.

I don't think the idea actually needs "saving" , so we disagree about that (and, it seems, about the results of the previous exchange), and I was trying to put it in a way that might be hopefully persuasive to you. So, that did not work, but I was not trying to persuade Speakpigeon of anything - I've already reckoned that there is no point in trying -, and while the thread is mostly meant to show in yet another way the untenability of Speakpigeon's positions, I was talking to you in that reply, not trying to convince readers of the failure of Speakpigeon's position.

When it comes to Speakpigeon's position, I do not think that the problem would be to step out of the formalism, but if there is a problem, the problem would be to step inside it in the first place - i.e., to use the formalism at all -, since Speakpigeon rejects the formalism altogether.

In any case, also when it comes to Speakpigeon's position, I started this thread not to interpret things in a first-order context, but rather, to show a case in which a contradiction is derived in the context of a standard mathematical argument, from an inconsistent set of premises. If they're not called premises, then from an inconsistent set of statements a contradiction is derived. In that context, the argument succeeded already - well, actually, it was not even necessary, since a similar argument succeeded in the other thread, but this one works too, for the following reason: In reply to my points, Speakpigeon made a distinction between an inconsistent set of premises and contradictory premises:

https://talkfreethought.org/showthr...Squid-Argument&p=694447&viewfull=1#post694447

One derives a contradiction.

The right word is indeed "inconsistent". The premises are inconsistent, i.e. one premise implies the negation of the other premise.

Contradictory premises would be p and not p and that's not what we have here.

The premises here are not contradictory.

You need to make sure you know the basics before posting silly arguments.
EB

So, Speakpigeon does not raise the same objections that you do (Speakpigeon clearly misunderstands what you're saying), and concedes that this - and similar cases - is an argument with inconsistent premises, but somehow claims that it is not a problem when "one premise implies the negation of the other premise.", but rather, the problem is with premises like "p and not p". Of course, that already debunks Speakpigeon's own claim that the "Improved Squid Argument" (and several similar arguments Speakpigeon asks about) is invalid, since that is precisely a case in which no premise contradicts itself, but one or more premises imply the negation of another.

So, this already worked! (by "worked" of course, I mean it worked in exposing Speakpigeon's position as absurd, not that it would do anything to persuade Speakpigeon, which I would not try).

As a final remark, and to throw in some real philosophy of mathematics, how does a logicist interpret your proof? They take logical validity to be powerful enough to cover all the mathematics we care about, and can show in their logics that arithmetical facts are just a subset of logical truths. All the ambient mathematical facts are now logical truths, and your proof shows that the statement that the square root of 2 is rational is analytically inconsistent with itself!

I'm not sure I get the terminology, but if I'm reading this right, then it seems under this view of the philosophy of mathematics, this is even better, for the following reasons:

1. This would be an example of a standard mathematical argument with an inconsistent set of premises. Sure, the set's cardinality would be 1, and the only premise would be, as you say, the premise that the square root of two is rational. But then, as you say, the proof under this interpretation shows that the statement the square root of 2 is rational is inconsistent with itself.

2. In addition to debunking Speakpigeon's position for the reason I explained above in this post, it would debunk Speakpigeon's position in yet another way: Speakpigeon accepts the proof as valid, but claims that arguments with contradictory premises are not valid. But alas, this is an argument with one contradictory premise, it seems.

That said, there might be a difficulty, not with regards to debunking Speakpigeon's position - that's already done -, but with regard to points 1. and 2. above (which is why I mentioned that I'm not sure I get the terminology): maybe your "analytically" qualifier makes a distinction between "inconsistent" and "analytically inconsistent", so I would like to ask whether you think that that would prevent 1. and/or 2. above (and if so, I'd like to ask for more details). My impression is that the "analytically" qualifier probably causes no trouble for either 1. or 2. above, since the logicist seems to consider analytical truths to be logical truths as you say (personally, I make a distinction between analytic truths and logical truths. For example, 'No bachelors are married' would be an example of the former, but not the latter, whereas any instance of a logically valid formula (e.g., ( P v ¬P)) would be an example of the latter).