Put quite clearly, this thread is NOT saying that the commutative property (in mathematics) is not real. I'm simply challenging anyone to prove it.
Let me phrase the question - Why should we accept the commutative property beyond the shadow of a doubt? Why should we think that it should be true of all numbers?
Just because it works a few times does not mean it is necessarily true for all numbers, so examples prove nothing.
My challenge - prove that a+b = b+a for any number a and b.
I just did that on my calculator, its 381460+430289i both ways.
The point I'm trying to get across is that actual examples don't prove anything. Why should we, based off of a few examples, make something an unshakeable rule for all of mathematics?
I took some interest in this and decided to search around for an answer. All proofs of the commutativity of addition that I see uses mathematical induction, which apparently shouldn't be confused with inductive reasoning and is a form of deductive reasoning(http://en.wikipedia.org/wiki/Mathematical_induction).
As far as it goes;
From the definition of addition, as the proof extensively uses the definition; s(a) = a + 1[A1] a + s(b) = s(a + b)[A2]
It is first necessary to prove that; a + s(b) = s(b) + a for all cases of a and b.
Proving that a + s(1) = s(a) + 1 using the definition of addition; a + s(1) = s(a + 1) using A2 s(a + 1) = s(s(a)) using A1 s(s(a)) = s(a) + 1 using A1
Proving that a + s(b) = s(a) + b; a + s(s(b)) = s(a + s(b)) using A2 s(a + s(b)) = s(s(a) + b) by induction s(s(a) + b) = s(a) + s(b) using A2 s(a) + s(b) = s(a) + b + 1 using A1 s(a) + b + 1 = s(s(a)) + b using A1 therefore, a + s(b) = s(a) + b[A3]
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Proving that a + 1 = 1 + a; s(a) + 1 = a + s(1) using A3 a + s(1) = s(a + 1) using A2 s(a + 1) = s(1 + a) by induction s(1 + a) = 1 + s(a) using A2 therefore, a + 1 = 1 + a
Proving that a + b = b + a; a + s(b) = s(a + b) using A2 s(a + b) = s(b + a) by induction s(b + a) = b + s(a) using A2 b + s(a) = s(b) + a using A3 therefore a + b = b + a --- Links: http://en.wikipedia.org/wiki/Proofs_involving_the_addition_of_natural_numbers http://www.dpmms.cam.ac.uk/~wtg10/addcomm.html --- I donât claim to be knowledgeable about the subject but I honestly have read and took the time to understand the proof I have given, so I didnât just copy and paste it. It took my interest so I thought I would look up on it, and honestly I am wondering a little if I chose the correct career path If you have problems with using induction then I cannot argue with you but perhaps you can try to take it up with a real mathematician and ask them why mathematical induction is considered a form of deductive proof, although Iâm not sure you will find such a person here on AG.
Can anyone here prove it, though? Or is it an assumption - an axiom?
I've posted what I've learned in page 2, and as far as I know the commutative property of addition can be proven by using the axiom of induction as a basis.
I took some interest in this and decided to search around for an answer. All proofs of the commutativity of addition that I see uses mathematical induction, which apparently shouldn't be confused with inductive reasoning and is a form of deductive reasoning
No - mathematical induction definitely should not be confused with inductive reasoning.
I think that the proof is very cool and elegant.
+1 to driejen
Along the way, I was expecting someone to post something like: a+b=b+a a+b-b=b+a-b a=a
But you see, by taking away b, you're assuming an axiom, because we are assuming that it doesn't matter which "b" you're taking away from a+b.
Now, what about the commutative property of multiplication? ab=a+a+a... "b times" ba=b+b+b... "a times"
So - when doing the proof, you have to think along the lines of the a+a+a... "b times"...
Why am I saying this? To make sure no one tries dividing ab=ba ab / a = ba / a b=b
because you get 1+1+1+1+1... "b times" and b/a + b/a + b/a ... "a times" - and this is even assuming the distribution property.
Perhaps you can use a similar method - induction - to show that 1+1+1+1... "b times" is the same as b "1 time" - then extend it for 2, 3, and so on, and all real numbers, and etc.
