How to prove or disprove algebraic equivalence?

[SFNQuestions]

I saw a discussion that got me wondering about problems related to abstract algebra and group theory. How you formally use group theory to disprove 2=3?

[Country Boy]

What makes you think that group theory has anything to do with that? The only things involved in "proving 2 is not equal to 3" are your definition of "2", your definition of "3", and your definition of "=". What are your definitions of those?

Edited March 14, 2017 by Country Boy

[SFNQuestions]

I could be wrong but I think the context is either within algebra or set theory. To me, you can look at this statement 2=3 and just see that it's wrong, but on the other hand, it's also often taken for granted that proving 1+1=2 takes pages of proof, so what's the actual proof behind proving it's wrong?

Edited March 14, 2017 by SFNQuestions

[SFNQuestions]

Wow a site full of "mathematicians" and no one knows, that's...kind of amazing, this should be a millennium prize problem I guess.

[wtf]

In set theory [math]2 = \{0,1\}[/math] and [math]3 = \{0,1,2\}[/math] and these are distinct sets by the axiom of extensionality. https://en.wikipedia.org/wiki/Axiom_of_extensionality

 

On the other hand in the Peano axioms, [math]2 \neq 3[/math] since two numbers are the same if and only if they are both successors of the same number; but [math]2 = S(1)[/math] and [math]3 = S(2)[/math].

 

https://en.wikipedia.org/wiki/Peano_axioms

 

Set theory and the Peano axioms are related by the Axiom of Infinity, which says (in effect) that there is a set that's a model of the Peano axioms. https://en.wikipedia.org/wiki/Axiom_of_infinity

Edited March 15, 2017 by wtf

[SFNQuestions]

In set theory [math]2 = \{0,1\}[/math] and [math]3 = \{0,1,2\}[/math] and these are distinct sets by the axiom of extensionality. https://en.wikipedia.org/wiki/Axiom_of_extensionality

 

On the other hand in the Peano axioms, [math]2 \neq 3[/math] since two numbers are the same if and only if they are both successors of the same number; but [math]2 = S(1)[/math] and [math]3 = S(2)[/math].

 

https://en.wikipedia.org/wiki/Peano_axioms

 

Set theory and the Peano axioms are related by the Axiom of Infinity, which says (in effect) that there is a set that's a model of the Peano axioms. https://en.wikipedia.org/wiki/Axiom_of_infinity

Alright, that makes sense. What if their difference is not equal to zero? Does that qualify as a proof?

[wtf]

Alright, that makes sense. What if their difference is not equal to zero? Does that qualify as a proof?

Yes that works too, directly from the order properties of the natural numbers or the reals. That's yet another proof.

Edited March 15, 2017 by wtf

[SFNQuestions]

Yes that works too, directly from the order properties of the natural numbers or the reals. That's yet another proof.

So what actually distinguishes this from a proof and just something you can look at that's taken to be self evident? A proof...does what? Breaks something down until only the axioms that define the particular group remain?

Edited March 15, 2017 by SFNQuestions

[wtf]

So what actually distinguishes this from a proof and just something you can look at that's taken to be self evident? A proof...does what? Breaks something down until only the axioms that define the particular group remain?

A proof is a logical deduction from some axioms. If you assume the Peano axioms you a "proof in PA" as they call it, and if you assume the axioms of set theory you get "proof in ZF".

 

I think in the present case the most fundamental proof is the one from set theory. The Peano axioms work but how do we know there is any model of them? So if you believe in the natural numbers then you can do a proof from Peano, but if you are a skeptic then you need the axiom of infinity to provide a model of them. That's a personal opinion. Most people accept PA as having some ontological referent, the "natural numbers of our intuition" or some such.

 

In the end it's just symbolic manipulation so you always have to start by taking some statements as given and not subject to proof.

 

So I guess you could say that [math]2 \neq 3[/math] is "obviously" true about the world. But in formal math, it's pretty arbitrary. The only reason [math]2[/math] and [math]3[/math] are different is that we define them that way. The symbol [math]3[/math] is defined as the successor of [math]2[/math], which is defined as the successor of [math]1[/math], which is the successor of [math]0[/math]. None of it means anything at all, it's just a symbolic game. That's the downside of formalization, you lose touch with reality. A child knows what it takes a logician years to prove.

Edited March 16, 2017 by wtf

[SFNQuestions]

A proof is a logical deduction from some axioms. If you assume the Peano axioms you a "proof in PA" as they call it, and if you assume the axioms of set theory you get "proof in ZF".

 

I think in the present case the most fundamental proof is the one from set theory. The Peano axioms work but how do we know there is any model of them? So if you believe in the natural numbers then you can do a proof from Peano, but if you are a skeptic then you need the axiom of infinity to provide a model of them. That's a personal opinion. Most people accept PA as having some ontological referent, the "natural numbers of our intuition" or some such.

 

In the end it's just symbolic manipulation so you always have to start by taking some statements as given and not subject to proof.

 

So I guess you could say that [math]2 \neq 3[/math] is "obviously" true about the world. But in formal math, it's pretty arbitrary. The only reason [math]2[/math] and [math]3[/math] are different is that we define them that way. The symbol [math]3[/math] is defined as the successor of [math]2[/math], which is defined as the successor of [math]1[/math], which is the successor of [math]0[/math]. None of it means anything at all, it's just a symbolic game. That's the downside of formalization, you lose touch with reality. A child knows what it takes a logician years to prove.

Well symbols aside, logic is logic regardless of whether we discover it or not. All logical correlations are true all the time. What if a particular set of axioms lacked a specific statement of uniqueness? Could you still prove it?

Edited March 16, 2017 by SFNQuestions

[wtf]

Well symbols aside, logic is logic regardless of whether we discover it or not.All logical correlations are true all the time.

Personally I'm not convinced about that. Did the plays of Shakespeare exist before Shakespeare wrote them? Before there were humans on earth?

 

Why would math or logic be different? There's ancient logic and modern logic. Modal logic, paraconsistent logic, all the modern stuff people have come up with. Where did these things exist before someone created/discovered/invented them?

 

I don't claim to know the answer.I don't think the latest Star Wars movie existed before someone wrote the script, hired the actors, and made the movie. Math and logic too.

 

 

What if a particular set of axioms lacked a specific statement of uniqueness? Could you still prove it?

I don't know what that means. Before Giuseppe Peano wrote down the axioms for the natural numbers, the world had no such axioms. If he'd written down different ones, history would be different. Many people believe the natural numbers have some sort of existence before there were humans. I'm not so sure. It's a matter of philosophy.

Edited March 16, 2017 by wtf

[wtf]

ps -- As it happens there is a thread on math.stackexchange today, where someone asks how to prove that two natural numbers are equal if and only if there's a bijection between them as sets. You can see from the responses that this no trivial matter!

 

http://math.stackexchange.com/questions/2189871/let-n-m-in-mathbbn-show-that-if-there-is-a-bijection-between-1-n-a?noredirect=1#comment4505167_2189871

Edited March 17, 2017 by wtf

[SFNQuestions]

Personally I'm not convinced about that. Did the plays of Shakespeare exist before Shakespeare wrote them? Before there were humans on earth?

 

Logically speaking, logic itself is not dependent on any amount of time nor any amount or space. Even if the universe didn't exist, 1+1=2 would still be a true statement relative to some immaterial set of axioms. But Shakespeare's play requires a physical universe to exist based on its very definition.

 

I don't know what that means. Before Giuseppe Peano wrote down the axioms for the natural numbers, the world had no such axioms. If he'd written down different ones, history would be different. Many people believe the natural numbers have some sort of existence before there were humans. I'm not so sure. It's a matter of philosophy.

That's true to some extent, but what I am referring to is ruling out the possibility that one value is equal to another value, just because you write different symbols in this "symbolic game." It seems like without uniqueness, 2=3 could actually be a true statement. And as you can imagine, even if an element of a set is succeeded by another element of a set, it doesn't seem like you could garuntee the values are unique when dealing with complex or imaginary numbers, or even just angles. You can't even say that one complex imaginary number is greater than another.

Edited March 17, 2017 by SFNQuestions

[wtf]

Logically speaking, logic itself is not dependent on any amount of time nor any amount or space. Even if the universe didn't exist, 1+1=2 would still be a true statement relative to some immaterial set of axioms.

Where do these axioms exist? I've seen many people argue that 1 + 1 = 2 in the absence of minds. But I don't think I've ever seen anyone claim that axiomatic systems themselves exist in the absence of people.

 

You are quite the Platonist. I am afraid I'm not that much of a believer. Different people believe different things about what's out there in some imaginary realm of nonphysical things. The natural numbers, the baby Jesus, the Flying Spaghetti Monster. That's my problem with Platonism. Once you start believing in the existence outside the mind of things that do not exist in the physical world ... where exactly do those things exist, and what else is living there?

 

 

But Shakespeare's play requires a physical universe to exist based on its very definition.

Yes if you define a play as something written down and acted out on a stage. But math in that sense doesn't exist till a professor writes it down and gets it accepted in a journal. You claim the math already existed and the play didn't. That's a matter of opinion, not fact.

 

And there are so many levels of this philosophical confusion. Before Wiles proved Fermat's last theorem, was FLT true? Even if a Platonist says that the theorem was ALWAYS true, did Wiles's particular proof already exist? Isn't the proof actually a physical thing written on paper by a human? It's no different than one of Shakespeare's plays. "A proof by any other name ..."

 

Of course my philosophical opinions are my own, so it's better if you ask me about the math. Everyone's got philosophical opinions.

 

 

That's true to some extent, but what I am referring to is ruling out the possibility that one value is equal to another value, just because you write different symbols in this "symbolic game." It seems like without uniqueness, 2=3 could actually be a true statement. And as you can imagine, even if an element of a set is succeeded by another element of a set, it doesn't seem like you could garuntee the values are unique when dealing with complex or imaginary numbers, or even just angles.

 

I can not exactly see what point you are trying to make. Can you state your thesis more clearly? Surely if you define the symbols differently you can make 2 and 3 mean the same thing. I don't see your core point here.

 

You can't even say that one complex imaginary number is greater than another.

As far as the complex numbers, there is no order on them compatible with their arithmetic. That's a theorem. When you jump up another level to the quaternions you lose commutativity, and when you go up to the octonions you even lose associativity.

 

I'm not sure I believe there's any deep philosophical meaning to any of this. Do you think there is?

 

Besides, even though the complex numbers lose order, they have one great improvement over the reals: they're algebraically complete. So you can't say the complex numbers are defective in any way, in fact they're much better than the reals for a lot of things.

Edited March 18, 2017 by wtf

[SFNQuestions]

I am afraid I'm not that much of a believer.

Well you're in luck then because there's nothing to believe. Where does the statement 1+1=2 say anything about time or space? It's not a question of whether or not it physically exists, it's just a matter of whether or not a logical correlation is true, and every one is true regardless of any position of anything or any amount of time that passes. No matter where or when you are in the universe, 1+1=2 is a true statement within any set of axioms that allow that statement to be true. For something closer to home, take for instance a dimension: we can't pick it up, we can't see one, but it is nonetheless the consensus in physics that dimensions exist.

 

 

I can not exactly see what point you are trying to make. Can you state your thesis more clearly? Surely if you define the symbols differently you can make 2 and 3 mean the same thing. I don't see your core point here.

As you said, a lack of defined natural numbers would imply a lack of uniqueness. What I'm asking is that if you defined operators like succession but didn't define uniqueness, it seems like there would be nothing to verify that 2=3 is false, even in your point that you could disprove the statement with succession since that relies on mathematics that has already been established. So the symbols 2 and 3 could be equal if there's nothing explicitly to say that any given number x and y (or a and b) have to be unique values. This isn't philosophy this is more on the main topic.

Edited March 18, 2017 by SFNQuestions

[wtf]

Well you're in luck then because there's nothing to believe. Where does the statement 1+1=2 say anything about time or space? It's not a question of whether or not it physically exists, it's just a matter of whether or not a logical correlation is true, and every one is true regardless of any position of anything or any amount of time that passes. No matter where or when you are in the universe, 1+1=2 is a true statement within any set of axioms that allow that statement to be true.

I think you are claiming two separate things, one much less believable than the other.

 

One, that 1 + 1 = 2 in the absence of any sentient minds in the universe to be aware of this fact. That is a Platonist position, but when I pretend to completely deny it, a little voice in me sort of believes it's true. I'm 5% Platonist.

 

On the other hand you are also claiming that some particular axiom system exists in the absence of a sentient mind. That seems to me less likely. I think we might make the fair distinction that even if 1 + 1 = 2 is a truth about the universe, the Peano axioms or the ZFC axioms of set theory are historically contingent works of man.

 

But twice now you have said that it's the axiom system that makes 1 + 1 true. But if you think about it, the axiom system is just a human invention to symbolically model what's already true in the universe, that 1 + 1 = 2.

 

Do you agree with me that axiom systems are contingent? And not ontologically on a par with the fact that 1 + 1 = 2?

 

 

 

For something closer to home, take for instance a dimension: we can't pick it up, we can't see one, but it is nonetheless the consensus in physics that dimensions exist.

Yes well it's the same point again. There's the Platonic physics, the true laws of nature. And there's human physics, from Aristotle to Newton and Einstein. Historically contingent physics, a mere approximation to the "true" physics.

 

And what makes you so certain there is any such thing as a true physics? Maybe it's all random and we just make the patterns up, like seeing constellations in the stars?

 

You see you are making an assumption and you don't quite see that you are making an assumption.

 

As you said, a lack of defined natural numbers would imply a lack of uniqueness. What I'm asking is that if you defined operators like succession but didn't define uniqueness, it seems like there would be nothing to verify that 2=3 is false, even in your point that you could disprove the statement with succession since that relies on mathematics that has already been established. So the symbols 2 and 3 could be equal if there's nothing explicitly to say that any given number x and y (or a and b) have to be unique values. This isn't philosophy this is more on the main topic.

If you had some formal system where the symbols 2 and 3 weren't necessarily unequal, then that's how it would be. If your system didn't have a rule that made them different, then in the formal system they would not be different.

 

Is that what you're trying to say? It doesn't seem very significant. If you define the symbols differently they mean something different.

 

And such a symbolic system wouldn't be very useful, because we're trying to model our intuition that 2 and 3 are different.

 

Now as far as WHY we have this intuition? Well a Platonist would say that it's because 2 and 3 really are different in the world. A non-Platonist would say, well I guess we just made it up. It's just as hard to be a non-Platonist as it is to be a Platonist!

Edited March 18, 2017 by wtf

[Country Boy]

Wow a site full of "mathematicians" and no one knows, that's...kind of amazing, this should be a millennium prize problem I guess.

 

No, it shouldn't. It is very easy to disprove "2= 3". But the question asked was "How you formally use group theory to disprove 2=3?" and the answer is "you don't". That is much more basic than "group theory"- it would be proved long before you got a complicated as group theory!