1x1=2

[Freodin]

If you're going to question what = means, then you'd might as well question what 'axiom' means. Or we can just agree that our knowledge of these concepts is sufficient to determine what's true.

We are talking mathematics here. In mathematics, you are free to ask what "=" means, and tinker with it. Read the linked wiki article. Gives some nice examples of what can and what cannot constitute "equivalence".

 

[Chriliman]

Oh, no, that is the point that you still have to grasp.
Variables / symbols are just placeholders for concepts. They are not inherently connected to what value is given to them.
In many cases, it might even be necessary to give identical values to them.

Another example: a classic binomical function.

(a+b)^2 = a^2 + 2*a*b + b^2

This is true for every value of a and b. Every value... even if a and b are identical.

Wouldn't you say that it is rather unnecessary to add to this formula: "In case that a and b are identical, the result it 4*a^2"?


That's the whole point. Variables can be filled with any value you want (within the given limits). These values could be identical... but they don't have to.

If the values are identical then using two symbols to represent the same value is redundant. You're more than welcome to use the same value for as many different symbols as you want in the same equation, but it's still redundant.

 

[Freodin]

If the values are identical then using two symbols to represent the same value is redundant. You're more than welcome to use the same value for as many different symbols as you want in the same equation, but it's still redundant.

No, it is not. The usage of seperate variables provide an explicit expression, that is valid regardless of the values used.

Take the triangle example again. A triangle always has three side. These three sides can be of different length. We name them differently: a, b and c.

Now in the cases where two or even all three of these variables hold the same value... all the expressions using three different variables still apply... and we need to use all three.

The triagle doesn't happen to now suddenly only have two or even one side, because the others became "redundant".
They are not. They are still there, still seperate concepts.

 

[Chriliman]

No, it is not. The usage of seperate variables provide an explicit expression, that is valid regardless of the values used.

Take the triangle example again. A triangle always has three side. These three sides can be of different length. We name them differently: a, b and c.

Now in the cases where two or even all three of these variables hold the same value... all the expressions using three different variables still apply... and we need to use all three.

The triagle doesn't happen to now suddenly only have two or even one side, because the others became "redundant".
They are not. They are still there, still seperate concepts.

Okay, in the case of the triangle we need to use separate variables because we don't know the values of the sides. If we knew the values of the sides then we'd just use the correct value to describe it, not the arbitrary symbols.

 

[Freodin]

Okay, in the case of the triangle we need to use separate variables because we don't know the values of the sides. If we knew the values of the sides then we'd just use the correct value to describe it, not the arbitrary symbols.

The "arbitrary symbols" are not arbitrary. They are given to the relevant concepts. It is the values that are arbitrary... we do not know them... that is the reason why we use symbols at all.

Perhaps you are getting to hung up with the example the Sultan of Swing presented. Forget it. It is not relevant. It is meant - it is designed! - to confuse people who haven't got the hang of algebra.

In this case, yes, the use of two variables is redundant. That is the point. This is the sleight of hand, the magician's diversion, the trick to keep the viewer from noticing what is happening.

 

[Chriliman]

The "arbitrary symbols" are not arbitrary. They are given to the relevant concepts. It is the values that are arbitrary... we do not know them... that is the reason why we use symbols at all.

Perhaps you are getting to hung up with the example the Sultan of Swing presented. Forget it. It is not relevant. It is meant - it is designed! - to confuse people who haven't got the hang of algebra.

In this case, yes, the use of two variables is redundant. That is the point. This is the sleight of hand, the magician's diversion, the trick to keep the viewer from noticing what is happening.

Right, and what I'm designed to do is seek out the truth and sometimes that means expose what's false.

 

[Freodin]

Right, and what I'm designed to do is seek out the truth and sometimes that means expose what's false.

But you didn't do that in this case. So why do you think you need to bring up your record in exposing what's false... when in this case you pointed out something that is correct as false, but ignored the really false thing?

 

[Chriliman]

But you didn't do that in this case. So why do you think you need to bring up your record in exposing what's false... when in this case you pointed out something that is correct as false, but ignored the really false thing?

?

I correctly pointed out that using two variables to describe the same value is illogical at post #48. I suppose I should have used the word redundant or unnecessary, instead of illogical, but you get my point.

 

[Freodin]

?

I correctly pointed out that using two variables to describe the same value is illogical at post #48. I suppose I should have used the word redundant or unnecessary, instead of illogical, but you get my point.

But that is not the problem with this skit.

So you think that the "redundancy" is what makes it wrong?

Let's do it again, then!

a = a (Axiom or not, it is correct)
a^2 = a^2 (Multiplying both sides of an equation with the same factor doesn't change the equality, so correct)
a^2 - a^2 = 0 (Subtracting the same value on both sides... still correct)
(a-a)(a+a) = 0 (valid transformation according the the binomic formulas... correct)
(a-a)(a+a)/(a-a) = 0/(a-a) (Dividing both sides with the same value... correct)
1(a+a) = 0 (reducing a fraction by eliminating equal factors in counter and denominator on the left side, using the fact that 0 multiplied by any fraction stays 0 on the right side of the equation... both correct.)
(a+a) = 0 (simplifying by just dropping the factor of one... correct)
1 + 1 = 0 (inserting a value... correct)
2 = 0 (computing the values... correct)
1 = 0 (not sure what was done here... division by 2? Still, correctly done.)
1 + 1 = 1 (using the last equation within a new calculation... if we were correct up to now, we can do this. So... correct.)

So, no redundancy anymore. Just one variable used.

Where is the problem?

 

[Chriliman]

1(a+a) = 0 (reducing a fraction by eliminating equal factors in counter and denominator on the left side, using the fact that 0 multiplied by any fraction stays 0 on the right side of the equation... both correct.)

I see a problem here. Doesn't 1(a+a) mean 1 times 1+1? Which would be 2.

(a+a) = 0 (simplifying by just dropping the factor of one... correct)

I see a problem here also. Why can you drop the 1 without showing subtraction?

 

[Freodin]

I see a problem here. Doesn't 1(a+a) mean 1 times 1+1? Which would be 2.

No, it wouldn't. Not right here. We haven't introduced any values yet. a could be anything.
Only later do we attribute 1 for a. But there is no need to do that... it just makes the trick more appealing. We could as well set a as 1155... we would run into the same problem, and a similar contradicting "2310 = 0" later.

I see a problem here also. Why can you drop the 1 without showing subtraction?

I spelled it out. This is a multiplication with a factor of 1. We have already found out that something multiplied with 1 stays the same. So we can simple leave it.
Subtraction on the other hand would be an error. There is no sum including a summand of 1 anywhere there. You wouldn't manage to get rid of the "1" on the left side of the equation... and to keep the balance, you would have to subtract 1 also from the right side. So we wouldn't have gained anything.


Come on, Chriliman! I already pointed out the flaw on the first go. You can just go back to the original posting of this and find my response to it. And then hopefully either understand what I pointed out as a flaw... or ask me to explain it.

 

[Chriliman]

No, it wouldn't. Not right here. We haven't introduced any values yet. a could be anything.
Only later do we attribute 1 for a. But there is no need to do that... it just makes the trick more appealing. We could as well set a as 1155... we would run into the same problem, and a similar contradicting "2310 = 0" later.


I spelled it out. This is a multiplication with a factor of 1. We have already found out that something multiplied with 1 stays the same. So we can simple leave it.
Subtraction on the other hand would be an error. There is no sum including a summand of 1 anywhere there. You wouldn't manage to get rid of the "1" on the left side of the equation... and to keep the balance, you would have to subtract 1 also from the right side. So we wouldn't have gained anything.


Come on, Chriliman! I already pointed out the flaw on the first go. You can just go back to the original posting of this and find my response to it. And then hopefully either understand what I pointed out as a flaw... or ask me to explain it.

Okay, I'll assume what you said earlier was correct even though I don't fully understand it :/

I don't like this slight of hand trickery that is called Maths.

 

[leftrightleftrightleft]

Mathematics as a system is based on a few unprovable assumptions or axioms.

Indeed.

Such assumptions and axioms include the symbology of numbers representing quantities and the fact that the set of natural numbers exist in a set of ever-increasing symbols which represent ever increasing quantities.

Amongst these are 1 +1 =2; 1 x 1 = 1 etc. They are axioms

No they aren't. They logically follow from axioms.

If we agree on the symbology (such as "+" means "to add together") and we also agree on the axioms ("1" represents a single quantity and "2" is the symbol to represent the number which follows "1"), then "1+1 = 2" logically follows as true.

It is not axiomatic. It logically follows from the definitions and axioms.

Even making statements like "you have 1 and add 1 to make 2" remain axiomatic. Why not get 3 or 715.21?

Because, given the set of axioms about how we understand the set of integer numbers, "1+1 = 715.21" does not logically follow unless we re-define the symbol "715.21" to mean a different quantity than it currently represents.

It has never been proven, only accepted as logical it is 2.

You are trying to philosophize this into oblivion. But it doesn't work.

You have one apple. You have another apple. How many apples do you have?

Do you have 715.21 apples?

Please.

So if someone alters the answer to a base axiom of Mathematics, he creates a fully acceptable variant Mathematics which is as valid as our normal one.

If the new system is not logically consistent, then it is useless. Not only that, it is not even mathematics because part of what makes math math is that it is logically consistent.

If I say "1x1 = Abraham Lincoln", have I invented a new math?

The axioms aren't proven as such and even if we disagree with his axiom, to HIM it is as plain as the nose on your face and remains axiomatically valid. Maybe the rest of us are wrong after all.

If this guy in the OP's article took like three minutes to try to reason his way through his new math, he would quickly discover that it is not logically consistent.

So while not correct in standard Mathematics, there is nothing inherently wrong about 1 x 1 = 2.

Only if you completely alter the meaning of what we agree are the definitions for the symbols "1", "2", and "x".

Currently the "a x b" (times) symbol means: "make a groups of b and add them together"

So 1 x 1 means: "make 1 group of 1 and add them together." You've only got one group of one so what is the total number of objects? 1.

How can he get 2? The only way he can do so is if he changes the definition of what multiplication means.

For example, he could say that "a x b" means: "multiply the numbers and add one"

In this case 1x1 = 2, 3x4 = 13, 2x2=5, etc. He has not invented a new math, he has just redefined what the symbol "x" means. It is no longer multiplication. It is something else.

In "True Mathematics" this might be the case, for there is no reason to assume that just because the vast majority of humanity agrees that 1 x 1= 1 is obviously true, that it MUST be.

It MUST be, based on our definitions of the symbols.

If someone wants to change the definitions, go for it. But they can no longer call it multiplication.

 

[Freodin]

Okay, I'll assume what you said earlier was correct even though I don't fully understand it :/

Quite easy.

It is not possible to divide by zero. The result is not defined within the systems we use in algebra.

You might be aware fractions can get very small, the bigger the denominator gets. 1/2 is smaller than 1. 1/3 is smaller than 1/2. 1/4 is smaller still. 1/10000000000000000 is already very small.

You might also be aware of the things you can do with fractions. As with all numbers, you can add, subtract, multiply... and divide... with fractions. When you divide by a fraction, the result gets bigger. Basically, what you do when you divide by a fraction is that you multiply with the reciprocal value. So 1 divided by 1/4 is equal to 1 multiplied by 4.

So now you could think that, if a fraction gets ever smaller, ever closer to zero, and a division by such an always smaller fraction gets bigger and bigger... wouldn't a division by the smallest of all - zero - be the biggest of all - infinity?

Yes, basically you would be correct. But there is a problem.

You might also be aware that there are such things as negative numbers. When you multiply or divide something positive with a negative number, the result gets negative.

So when you take a postive fraction to get ever closer to zero, and divide with it, the result will get bigger and bigger, up to infinity when you reach zero.
But when you take a negative fraction to get ever closer to zero, and divide with it, the result will get ever bigger and bigger... on the negative side. Up to -infinity when you reach zero.

So something divided by zero would be BOTH positive infinite AND negative infinite. That isn't possible. Something divided by zero results in a contradiction.


And this is what happens in this mathe-magic trick. We do divide by (a-a) - which is of course zero. And thus our result gets invalidated. It becomes a contradiction.

I don't like this slight of hand trickery that is called Maths.

This isn't maths. This is a trick played with maths. Maths works quite well, when you know what you are doing, and are aware of some things.

 

[FrumiousBandersnatch]

I correctly pointed out that using two variables to describe the same value is illogical at post #48. I suppose I should have used the word redundant or unnecessary, instead of illogical, but you get my point.

Variables are placeholders for values. You use different variables (a, b, c, x, y, etc) because the values in each can change (e.g. you can add a value to a variable so it contains a new value), and you generally want to see what happens to the values in the variables when you perform mathematical operations with them. You can also run the same set of mathematical operations several times with different values in the variables when you start, to see what happens.

 

[Quid est Veritas?]

No they aren't. They logically follow from axioms.

If we agree on the symbology (such as "+" means "to add together") and we also agree on the axioms ("1" represents a single quantity and "2" is the symbol to represent the number which follows "1"), then "1+1 = 2" logically follows as true.

It is not axiomatic. It logically follows from the definitions and axioms.

No, they are axioms. To assume that to 'add together' adds from 1 to 2 is axiomatic. Just because we defined something a certain way does not mean it has to follow. To assume it does, is an axiom. To say something 'logically follows' means you assume it is self-evident which by the definition of the term makes it an axiom even if a part of the axiom is itself an axiom.
But symbols aren't axioms as we need to be taught what they mean, nor are definitions.

Because, given the set of axioms about how we understand the set of integer numbers, "1+1 = 715.21" does not logically follow unless we re-define the symbol "715.21" to mean a different quantity than it currently represents.

Again, to say it doesn't 'logically follow' means you take it as an axiom that it does. It was never proven that 1 + 1 is 2. Please show me the mathematical proof that this is so if you do not consider it an axiom. Just because you can show a real world example where 1 apple and another one became 2 apples does not prove the principle is universal.
Else I could say that Pythagoras theorum 'logically follows' if I found it true in one triangle - for why would I need to supply a proof? Mathematics doesn't work like that.

You are trying to philosophize this into oblivion. But it doesn't work.

If the new system is not logically consistent, then it is useless. Not only that, it is not even mathematics because part of what makes math math is that it is logically consistent.

It may well be logically consistent, just not using what you yourself would consider logical.
Its the old story of the Tooth Fairy - if a child catches their parent switching the tooth for money, both thinking the tooth fairy is hocum or that their parent is the tooth fairy are logical deductions. Logic can be tricky.

Only if you completely alter the meaning of what we agree are the definitions for the symbols "1", "2", and "x".

Currently the "a x b" (times) symbol means: "make a groups of b and add them together"

So 1 x 1 means: "make 1 group of 1 and add them together." You've only got one group of one so what is the total number of objects? 1.

How can he get 2? The only way he can do so is if he changes the definition of what multiplication means.

For example, he could say that "a x b" means: "multiply the numbers and add one"

In this case 1x1 = 2, 3x4 = 13, 2x2=5, etc. He has not invented a new math, he has just redefined what the symbol "x" means. It is no longer multiplication. It is something else.




It MUST be, based on our definitions of the symbols.

If someone wants to change the definitions, go for it. But they can no longer call it multiplication.

He very much changed the definitions of things. This does not mean his system is any more wrong or right than ours. The two are mutually exclusive as their base axioms differ. But as Freodin and I already went over this at some length, I suggest you read our exchange. There is no reason for me to repeat myself and he made valid points which we discussed, to which I think you yourself are tending.

 

[essentialsaltes]

It was never proven that 1 + 1 is 2. Please show me the mathematical proof that this is so if you do not consider it an axiom.

I would refer you to Whitehead and Russell's Principia Mathematica, which sought to place mathematics upon a sound logical foundation. "Famously, several hundred pages are required in PM to prove the validity of the proposition 1+1=2."

 

[Quid est Veritas?]

I would refer you to Whitehead and Russell's Principia Mathematica, which sought to place mathematics upon a sound logical foundation. "Famously, several hundred pages are required in PM to prove the validity of the proposition 1+1=2."

Interesting, but he would still have started with some form of axiom. I shall look for the proof or more likely, someone who can explain it to me. Do you know if they attempted a proof of 1 x 1 = 1?

 

[Quid est Veritas?]

I would refer you to Whitehead and Russell's Principia Mathematica, which sought to place mathematics upon a sound logical foundation. "Famously, several hundred pages are required in PM to prove the validity of the proposition 1+1=2."

Apparently Kurt Godel showed in 1931 with two incompleteness theorums that all axiomatic systems containing Arithmetic are inherently limited. So even if the proof of that specific equation is found, its still founded on a system that is not consistent or complete in and of itself, nor can be. Apparently it was written directly as a response to the PM mentioned before.

Not very familiar with this myself though, so I shall have to investigate, but it makes sense philosophically.

 

[essentialsaltes]

Interesting, but he would still have started with some form of axiom.

You can't prove anything without a logical system, and a logical system needs axioms.