1x1=2

[Quid est Veritas?]

That is not quite correct.

You are correct though that all mathematical systems are based on unproven axioms, which are taken as "obvious truths". But you are not correct in identifying these axioms. Neither "1+1=2" nor "1x1=1" are axioms... they are definitions.

For example, there is a mathematical concept called "field" (I had to google that, in German it is called a "Körper"... a body ). A field is a bunch of elements (not even necessarily "numbers") and operations (usually called addition and multiplication, but not necessarily the "normal" kind) where certain rules apply. Like: "any operation done on an element of the field always results in an element of the field".

Another rule of a "field" is "there exists a "neutral" element for operations, where any operation done on one element with that neutral element results in the original element" Mathspeak: "x + n_add = x , x * n_mult =x"

The set of natural numbers form such a field. The neutral element for multiplikation is said to be "1". Thus 1 * 1 = 1.

"2" is just the definition for the element that follows "1" in the set of natural numbers.

But now there is an axiom: "for every element n in the set of natural numbers, there exist an element n+1" Unproven, unprovable... but obviously correct.

Mathematics is awesome.

Definition means a "statement of the exact meaning of a word or concept". How does this differ from an axiom, if your definition is unprovable?
To my mind, 1 x 1 = 1 is both a definition as well as an axiom. If our friend defines the value of 1 x 1 differently, it remains a valid system.

As to n always having a value n + 1 with the neutral element being 1: the assumption here that n + 1 is 2 if n is the neutral element 1, is of course itself axiomatic even if 2 is taken as the next element in the 'field' - why would addition yield the next element?

 

[Freodin]

Definition means a "statement of the exact meaning of a word or concept". How does this differ from an axiom, if your definition is unprovable?
To my mind, 1 x 1 = 1 is both a definition as well as an axiom. If our friend defines the value of 1 x 1 differently, it remains a valid system.

An axiom is hold to be true, even if it cannot be proven. A definition IS true, per definitionem.
If you define 1 * 1 (Sorry, I don't like to use x as sign for multiplication... too much confusion potential with x as variable... what is x multiplied with x? x x x ? ) differently as in the natural numbers ... of course you would be correct.

Whether you could build a usable mathematical system on it... that is another question. You would also redefine multiplication to get a valid system.
Yes, it can be done. But it would be a different system from the natural, rational, real number system.
It wouldn't get you any applause from the mathematical world... it is nothing new, already been done.
And it wouldn't get you any advantage in the "real" world... where natural numbers (and there rational and real extentions) work very well.
And if you insist that your system is correct, and all others are not... like the guy in the OP does... it only shows that you didn't understand the mathematics behind it.

As to n always having a value n + 1 with the neutral element being 1: the assumption here that n + 1 is 2 if n is the neutral element 1, is of course itself axiomatic even if 2 is taken as the next element in the 'field' - why would addition yield the next element?

The neutral element of addition would be 0. 1 is the neutral element for multiplication.

But as for your question... it is defined this way. The way "addition" is defined in the natural numbers is that adding the smallest possible not-neutral element gets you to the "next" element. What this element is called is rather irrelevant. But you can show that it has certain properties.

 

[Freodin]

...
The set of natural numbers form such a field. The neutral element for multiplikation is said to be "1". Thus 1 * 1 = 1.
...
Mathematics is awesome.

Mathematics is awesome... and sometimes a little complicated.

Of course the set of natural numbers do NOT form a field. For a set to be a field, it also needs to include "inverse" elements... so that "x + (x_add_inverse) = n_add" and "x * (x_mult_inverse) = n_mult".

As the natural numbers do not include all inverse elements for multiplication - the fractions - it isn't a field.
The set of rational and real numbers are a field though.



Math still is awesome.

 

[Chriliman]

No, no, no, please, Chriliman. Look back at what you wrote earlier.
"Always assume there is someone who knows more than you do and that you can learn something from them."

"a" and "b" are just two seperate variables. Two seperate expressions that can be filled with values, or assigned to seperate attributes. They do not depend on each other.

So simply saying "if a = 1, then b ≠ 1" is not "logical". Just as simply saying "if a = 1, then b = 1" is not "logical". There is not yet any relation established between a and b.
Such a relation needs to be established, either by definition or by calculation.

For example:

Let a be the number of dollar bills in your wallet. Let b be the price of bread in dollars. c will be the number of loafs of bread you can buy. c=a/b

Now to fill it with values. Assume you have ten dollars, and the price of bread is two dollars per loaf.

a=10, b=2
c=a/b
c=10/2
c=5
​

You can buy five loafs of bread.

Or you have one dollar, and the price of bread is one dollar per loaf.

a=1, b=1
c=a/b
c=1/1
c=1
​

Just one bread now. See, different variables... same value. In this case, even three variables with the same value! Got it?

Another example:
The famous "Pythagorean theorem": in any triangle with one right angle (90°), the square of the side opposite the right angle (the hypotenuse, called "c") is equal to the sum of the squares of the other two sides (the catheti, called "a" and "b").
You certainly have heard it or read it somewhere... "a squared plus b squared is c squared"

Example: Right angled triangle, the cathetus a = 3, the cathetus b = 4. What is the length of the hypotenuse?

c^2 = a^2 + b^2
c = sqr (a^2 + b^2)
In this case: c = sqr(3*3 + 4*4)
c = sqr(9+16)
c = sqr(25)
c = 5
​

Example: Right angled triangle, cathetus a = 1, hypothenuse c = sqr(2). What is the length of the other cathetus b?

Again: c^2 = a^2 + b^2
Thus: b^2 = c^2 - a^2
b = sqr(c^2 - a^2)
In this case: b = sqr( sqr(2)*sqr(2) - 1*1)
b = sqr(2 - 1)
b = sqr(1)
b = 1.... equal to what a is.
​

Different variables, denoting different attributes... same value.

It's not rocket science, or brain surgery. Just simple algebra.

Nice Maths!

Spot the axiom and contradiction:

1. If a=1, then 1≠b <axiom or not?

2. If a=1, then 1≠a <contradiction or not?

Again, spot the axiom and contradiction:

1. If a=1, then 1=b <contradiction or not?

2. If a=1, then 1=a <axiom or not?

Please answer every question accurately

 

[Chriliman]

If a = 1 and b = 1, then a = b because 1 = 1. That is completely and intrinsically logical.

-CryptoLutheran

I understand what you're saying. I just don't understand the purpose of using two different variables/symbols to represent the same value/meaning.

It would make more logical sense to say:

If a=1 and b=2, then a+b=3

It seems that if a=1 then 'a' cannot equal anything but 1 and 1 cannot equal anything but 'a', in order for the equation to remain logical and coherent. To say 'b' also equals 1, just seems redundant and unnecessary to me, but I do understand what your saying

 

[Quid est Veritas?]

An axiom is hold to be true, even if it cannot be proven. A definition IS true, per definitionem.

Not exactly. A definition is absolutely true if I apply that definition, but it might not be true at all otherwise. If our friend looks at our definitions, they aren't true in his estimation as they haven't been accepted as such. The aren't valid definitions in his opinion. So how do we establish the validity of our definitions? By axiomatically assuming they are valid as we defined them as such. It all boils down to Wittgenstein - the simplistically stated meaning is use to determine validity - we use them so they are true. If we do not, they are axioms as they would then not be a definition in our framework.
Language can also be awesome!

If you define 1 * 1 (Sorry, I don't like to use x as sign for multiplication... too much confusion potential with x as variable... what is x multiplied with x? x x x ? ) differently as in the natural numbers ... of course you would be correct.

Whether you could build a usable mathematical system on it... that is another question. You would also redefine multiplication to get a valid system.
Yes, it can be done. But it would be a different system from the natural, rational, real number system.
It wouldn't get you any applause from the mathematical world... it is nothing new, already been done.
And it wouldn't get you any advantage in the "real" world... where natural numbers (and there rational and real extentions) work very well.
And if you insist that your system is correct, and all others are not... like the guy in the OP does... it only shows that you didn't understand the mathematics behind it.

Our friend probably didn't understand it. Also his system implicitly does alter the definition of 1 x 1.
But he could turn around and say to us "You simply do not understand Mathematics" and he would be correct. We don't understand his mathematics.
While useless in the real world, its not philosophically speaking, less valid. An oddity perhaps, but not something I can prove unequivocally wrong as we disagree on the base meanings of terms.

The neutral element of addition would be 0. 1 is the neutral element for multiplication.

But as for your question... it is defined this way. The way "addition" is defined in the natural numbers is that adding the smallest possible not-neutral element gets you to the "next" element. What this element is called is rather irrelevant. But you can show that it has certain properties.

As far as I can tell, you can show it has certain properties only in relation to other elements? By preceding another say? You cannot show it has certain properties in and off itself, but only in relation. What property has 4 say? Only its four-ness. To say it comes before 5 or after 3 or is divisible by 2 are relational ideas which of course all rest on how we defined the relationships between the elements in the first place.
Our definition created the properties, they don't inherently exist.

 

[JacksBratt]

I wouldn't be surprised if the very foundation of mathematics is based on falsehood.

What does it mean to 'times' something? Take an amount and duplicate it by another amount. So if you take 1 and duplicate it 1 time, you have to 2. If you take 2 and duplicate it 2 times, you have 4. Makes sense to me.

Edit: now that I look at that again, if you have 2 and duplicate it 2 times you have 6. Something doesn't seem right there...

1 x 1 is not "duplicating" anything. It is saying you have 1 quantity of 1 which is 1. This is better described by saying 1 of 1.

2 of a quantity of 2 is 4, or 2 of 2.

Let me show you with a simple picture... here is an "X"
There is one "X"
We have 1 times (or amount) of 1 "X"


Here is two "X"'s XX

We have 1 times (amount of ) two "X"'s so the total is 2 "X"'s

Here is 2 times 2 "X"'s XX XX

We have two "X"'s but twice, or two times. We have two "X"'s for two occurrences. So we have four "X"'s.

It's not new math. It's twisting what is meant by the "times" sign.

If you had sixteen piles of rocks and 10 rocks in each pile you would describe this with the word equation of having sixteen times 10. Or 10 rocks sixteen times. Or sixteen occurrences of 10 rocks.

The original 1 x 1 would be better described as 1 pile of 1 rock. OR 1 rock 1 time. OR one occurrence of 1 rock.

 

[JackRT]

3X5=15 could be stated: "if I have three groups of five objects each, then I have fifteen objects"

1X5=5 could be stated: "if I have one group of five objects, then I have five objects"

1X1=1 could be stated: "if I have one group of one object, then I have one object"

1+1=2 could be stated: "if I have one object and add another to it then I have two objects"

 

[Armoured]

3X5=15 could be stated: "if I have three groups of five objects each, then I have fifteen objects"

1X5=5 could be stated: "if I have one group of five objects each, then I have five objects"

1X1=1 could be stated: "if I have one group of one object, then I have one object"

1+1=2 could be stated: "if I have one object and add another to it then I have two objects"

Not rocket surgery, you'd think.

 

[Chriliman]

This test is open to anyone to answer. I'm interested to see what people think.

Spot the axiom and contradiction:

1. If a=1, then 1≠b <axiom or not?

2. If a=1, then 1≠a <contradiction or not?

Again, spot the axiom and contradiction:

1. If a=1, then 1=b <contradiction or not?

2. If a=1, then 1=a <axiom or not?

 

[JacksBratt]

3X5=15 could be stated: "if I have three groups of five objects each, then I have fifteen objects"

1X5=5 could be stated: "if I have one group of five objects each, then I have five objects"

1X1=1 could be stated: "if I have one group of one object, then I have one object"

1+1=2 could be stated: "if I have one object and add another to it then I have two objects"

EXACTLY

We need to go back to 1st grade and review what each of the arithmetical signs actually mean in words.

 

[JackRT]

1. If a=1, then 1≠b <axiom or not? ----------------not

2. If a=1, then 1≠a <contradiction or not? -------- contradiction



1. If a=1, then 1=b <contradiction or not? --------- not a contradiction but the "then' is redundant

2. If a=1, then 1=a <axiom or not? ----------------- axiom

 

[Freodin]

Nice Maths!

Spot the axiom and contradiction:

1. If a=1, then 1≠b <axiom or not?

2. If a=1, then 1≠a <contradiction or not?

Again, spot the axiom and contradiction:

1. If a=1, then 1=b <contradiction or not?

2. If a=1, then 1=a <axiom or not?

Please answer every question accurately

Nice evasion, pal. There I go, write out a whole math lesson... and you don't react with a single word to show that you understood it, have problems with it that I could help you with... or that you even read it.
What was that about "assuming that other people may know more than you and that you could learn from them"?

But as I am me, and cannot change my inner-teacher-self, I will answer your questions.

1. "If a =1, then 1≠b"
No, not an axiom. This isn't an obvious truth that a mathematical system is build on. I would even say here, can be build on.
As I have shown to you - had you read it - I gave you examples with a and b where both were equal or were not equal.
Whether a equals or not-equals b depends on what a and b stand for.

2. "If a=1, then 1≠a"
Contradiction. No matter if it is "1≠a" or "a≠1"... that is what "equality" means: if one side equals (or not-equals) the other side, it doesn't matter which side you write first.

But when you say that one side both equals and not equals the other... then of course this is a contradiction. The "if" condition is quite meaningless in this context... it simply doesn't matter.

1. "If a=1, then 1=b"
No, not a contradiction. Just very dependent on the context. What kind of context did you have in mind, and why would you think this is a contradiction?

2. If a=1, then 1=a
I am not qualified enough to definitly say that this is an axiom, but as you asked, I would say: yes, this is one.
Equality means that you can replace one side with the other.
Thus a=1 and 1=a can both be written as "a=a" and "1=1".
I'd say that the identity of an element with itself is an obvious truth.

 

[essentialsaltes]

2. If a=1, then 1=a
I am not qualified enough to definitly say that this is an axiom, but as you asked, I would say: yes, this is one.

That's a particular example of the symmetric property of equality. These properties are found in all equivalence relations. I'm no expert either, but I would not say this is an axiom, rather it is a consequence of choosing the standard = to be the equivalence relation in standard arithmetic.

 

[Freodin]

Not exactly. A definition is absolutely true if I apply that definition, but it might not be true at all otherwise. If our friend looks at our definitions, they aren't true in his estimation as they haven't been accepted as such. The aren't valid definitions in his opinion. So how do we establish the validity of our definitions? By axiomatically assuming they are valid as we defined them as such. It all boils down to Wittgenstein - the simplistically stated meaning is use to determine validity - we use them so they are true. If we do not, they are axioms as they would then not be a definition in our framework.
Language can also be awesome!

Not exactly back to you!
It works a little different with mathematics. If you take your basic axioms for a system, you can use definitions to derive "meaning" from them.

So if you define something in mathematics... it IS true. The axoims relate to these definitions.

Of course, you can in this way define anything already established in maths differently. But that wouldn't be what others define as "mathematics" any more.

Our friend probably didn't understand it. Also his system implicitly does alter the definition of 1 x 1.
But he could turn around and say to us "You simply do not understand Mathematics" and he would be correct. We don't understand his mathematics.
While useless in the real world, its not philosophically speaking, less valid. An oddity perhaps, but not something I can prove unequivocally wrong as we disagree on the base meanings of terms.

It doesn't give objective truth, correct. If you don't want to stay within any part of the established system at all, you are free to do so. But that is the problem with the OP-guy: by stating that others are false in their interpretation, he shows that he does want to stay within this framework. And the framework itself shows that he is wrong.

As far as I can tell, you can show it has certain properties only in relation to other elements? By preceding another say? You cannot show it has certain properties in and off itself, but only in relation. What property has 4 say? Only its four-ness. To say it comes before 5 or after 3 or is divisible by 2 are relational ideas which of course all rest on how we defined the relationships between the elements in the first place.
Our definition created the properties, they don't inherently exist.

Again, not quite correct. Our definition create some of the properties. Others are necessarily derived from these, and thus exist inherently.

 

[Chriliman]

Nice evasion, pal. There I go, write out a whole math lesson... and you don't react with a single word to show that you understood it, have problems with it that I could help you with... or that you even read it.
What was that about "assuming that other people may know more than you and that you could learn from them"?

But as I am me, and cannot change my inner-teacher-self, I will answer your questions.

1. "If a =1, then 1≠b"
No, not an axiom. This isn't an obvious truth that a mathematical system is build on. I would even say here, can be build on.
As I have shown to you - had you read it - I gave you examples with a and b where both were equal or were not equal.
Whether a equals or not-equals b depends on what a and b stand for.

I think in this case it's unnecessary to say 1≠b because we can't possibly know that based on a=1.

2. "If a=1, then 1≠a"
Contradiction. No matter if it is "1≠a" or "a≠1"... that is what "equality" means: if one side equals (or not-equals) the other side, it doesn't matter which side you write first.

But when you say that one side both equals and not equals the other... then of course this is a contradiction. The "if" condition is quite meaningless in this context... it simply doesn't matter.

Agreed!

1. "If a=1, then 1=b"
No, not a contradiction. Just very dependent on the context. What kind of context did you have in mind, and why would you think this is a contradiction?

Again, I think in this case it's unnecessary to say 1=b if all we know is that a=1. This goes to my point that it's unnecessary to have two variables/symbols equal the same value/meaning as in a=1 and b=1.

2. If a=1, then 1=a
I am not qualified enough to definitly say that this is an axiom, but as you asked, I would say: yes, this is one.
Equality means that you can replace one side with the other.
Thus a=1 and 1=a can both be written as "a=a" and "1=1".
I'd say that the identity of an element with itself is an obvious truth.

Agreed!

 

[Quid est Veritas?]

Not exactly back to you!
It works a little different with mathematics. If you take your basic axioms for a system, you can use definitions to derive "meaning" from them.

So if you define something in mathematics... it IS true. The axoims relate to these definitions.

Of course, you can in this way define anything already established in maths differently. But that wouldn't be what others define as "mathematics" any more.

I stated I was using Axioms in the epistemologic sense, not the Mathematical one in my second post. This is a different meaning of the word entirely.

It doesn't give objective truth, correct. If you don't want to stay within any part of the established system at all, you are free to do so. But that is the problem with the OP-guy: by stating that others are false in their interpretation, he shows that he does want to stay within this framework. And the framework itself shows that he is wrong.

Touché. Yes, you have a point here.

Again, not quite correct. Our definition create some of the properties. Others are necessarily derived from these, and thus exist inherently.

I don't follow. A property derived from another property which was created by the definition, remains dependant on the initial definition and therefore has no inherent existence.

 

[Freodin]

I think in this case it's unnecessary to say 1≠b because we can't possibly know that based on a=1.

Again, I think in this case it's unnecessary to say 1=b if all we know is that a=1. This goes to my point that it's unnecessary to have two variables/symbols equal the same value/meaning as in a=1 and b=1.

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.

 

[Chriliman]

That's a particular example of the symmetric property of equality. These properties are found in all equivalence relations. I'm no expert either, but I would not say this is an axiom, rather it is a consequence of choosing the standard = to be the equivalence relation in standard arithmetic.

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.

 

[Freodin]

I stated I was using Axioms in the epistemologic sense, not the Mathematical one in my second post. This is a different meaning of the word entirely.

And a touché for you.

I don't follow. A property derived from another property which was created by the definition, remains dependant on the initial definition and therefore has no inherent existence.

They are not defined themselves, are they? So where does such a property come from?
I'd say that it is a necessary derivate from the definitions. Thus it is an inherent property of the existence as a denpedant element.