Science and Randomness

[The Grand Nudger]

Whether or not it's true has nothing to do with whether or not it's an axiom. There's no controversy, because it is a statement that follows logically -from- axioms.....if we're only containing it to a discussion of the system itself.

[Mark 13:13]

Whether or not it's true has nothing to do with whether or not it's an axiom. There's no controversy, because it is a statement that follows logically -from- axioms.....if we're only containing it to a discussion of the system itself.

so which axioms did 1+1=2 need to be defined from or follow logically from ? Does it not stand as an axiom now as does it not meet the definition of axiom, In which way does in match the definition of an axiom.

[The Grand Nudger]

Now I have to do your googling for you as well....

In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems.
http://en.wikipedia.org/wiki/Axiom

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

Perhaps these links will help you to understand why 1+1=2 is not an axiom (since my summarizing it in a few sentences posts ago did not).

[Mark 13:13]

Now I have to do your googling for you as well....

In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems.
http://en.wikipedia.org/wiki/Axiom

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

Perhaps these links will help you to understand why 1+1=2 is not an axiom (since my summarizing it in a few sentences posts ago did not).

we are at an impasse and before we continue I think we need more opinion so I will wait to see what others say on this.

[pocaracas]

I googled all the way to Peano axioms too, but I was on my mobile, so I'd take some half an hour to reply with that...
Let's see the important ones for the 1+1=2 discussion.

1. 0 is a natural number.

6. For every natural number n, S(n) is a natural number.

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1 and 6 define a unary representation of the natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)) (which is also S(1)), and, in general, any natural number n as Sn(0). The next two axioms define the properties of this representation.

7. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

[...]

Addition is the function + : N × N → N (written in the usual infix notation, mapping two elements of N to another element of N), defined recursively as:
a+0=a,
a+S(b) = S(a+b)

Now, do the math!

[Mark 13:13]

I googled all the way to Peano axioms too, but I was on my mobile, so I'd take some half an hour to reply with that...
Let's see the important ones for the 1+1=2 discussion.

1. 0 is a natural number.

6. For every natural number n, S(n) is a natural number.

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1 and 6 define a unary representation of the natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)) (which is also S(1)), and, in general, any natural number n as Sn(0). The next two axioms define the properties of this representation.

7. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

[...]

Addition is the function + : N × N → N (written in the usual infix notation, mapping two elements of N to another element of N), defined recursively as:
a+0=a,
a+S(b) = S(a+b)

Now, do the math!

hock: i know where you are at so i've tried to not get into anything to complicated, but knowing some of the members of the forum they will be able to settle the issue and explain it simply without getting too confusing.

[The Grand Nudger]

we are at an impasse and before we continue I think we need more opinion so I will wait to see what others say on this.

Why are we at an impasse, at no point, in responding to my gradually more elaborate responses have you ever taken the time to mention what parts you disagree (or agree) with.

[pocaracas]

I googled all the way to Peano axioms too, but I was on my mobile, so I'd take some half an hour to reply with that...
Let's see the important ones for the 1+1=2 discussion.

Now, do the math!

hock: i know where you are at so i've tried to not get into anything to complicated, but knowing some of the members of the forum they will be able to settle the issue and explain it simply without getting too confusing.

So I have to do the math for you?

1+1 =
(To replicate the nomenclature in the definition of addition)
= 1+ S(0) =
(Second part of the definition of addition)
= S(1+0) =
(1+0 = 1 as per the first part of the definition of addition)
= S(1) =
(According to axiom 6.)
= 2

[Mark 13:13]

hock: i know where you are at so i've tried to not get into anything to complicated, but knowing some of the members of the forum they will be able to settle the issue and explain it simply without getting too confusing.

So I have to do the math for you?

1+1 =
(To replicate the nomenclature in the definition of addition)
= 1+ S(0) =
(Second part of the definition of addition)
= S(1+0) =
(1+0 = 1 as per the first part of the definition of addition)
= S(1) =
(According to axiom 6.)
= 2

thanks for that.
i think i got it now ok so 1+1= 2 is not an axiom within that system. but mathematics is still based on axioms. So is still based on statements that are not provable using logic but accepted using other methods of decision be they intuition or common sense or whatever we deem to call it. (And maybe 1+1 =2 could as it now would appear to fit the above definition now be accepted in at least some systems as an axiom)

[The Grand Nudger]

As a system, sure. If we were doing addition with physical objects...however....we can use all the logic or evidence we like, no intuition required. The system helps us to organize and communicate what we're up to.