- #121
drcrabs
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learningphysics said:
Cheers bro
Is Addition Really a Basic Skill?
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The discussion centers on the nature of addition and whether it can be proven. Participants argue that addition is a fundamental concept defined by axioms rather than something that can be proven in the traditional sense. The Peano Axioms are referenced as a formal system that defines natural numbers and addition, illustrating that addition is inherently intuitive and based on human understanding of quantity. The conversation highlights that while one can demonstrate properties of addition (like commutativity), the operation itself is defined rather than proven. Some participants express frustration over the vagueness of the original question about proving addition, emphasizing that mathematical operations are accepted based on axiomatic foundations rather than empirical proof. The discussion also touches on the philosophical implications of mathematical definitions, particularly in relation to set theory and the concept of individuality in counting. Overall, the consensus is that addition is a defined operation rooted in axiomatic logic, not a provable theorem.
Cheers bro
- #122
matt grime
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It should also be said that in Hurkyl's field the polynomial x^2-2 splits, and hence indeed there is such a thing as the square root of 2.
As for the 10/7 1.4 thing, you implication was that you ought stop after two steps in the construction of the decimal expansion, and that because these two things aren't equal the square root of 2 doesn't exist in the real numbers. That is what is odd about your argument. I didn't say you claimed they were equal I claimed you implied that they *ought* to be equal, and that because they aren't sqrt(2) is self referenced (whatever that may actually mean) and therefore isn't a valid concept.
Are you going to answer the questions about self referencing and its implications?
note there is a real number whose square is 2 since R is complete and f(x)=x^2 is continuous f(0)=0, f(2)=4, thus the intermediate value theorem applies.
As for the 10/7 1.4 thing, you implication was that you ought stop after two steps in the construction of the decimal expansion, and that because these two things aren't equal the square root of 2 doesn't exist in the real numbers. That is what is odd about your argument. I didn't say you claimed they were equal I claimed you implied that they *ought* to be equal, and that because they aren't sqrt(2) is self referenced (whatever that may actually mean) and therefore isn't a valid concept.
Are you going to answer the questions about self referencing and its implications?
note there is a real number whose square is 2 since R is complete and f(x)=x^2 is continuous f(0)=0, f(2)=4, thus the intermediate value theorem applies.
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- #123
Hurkyl
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I remembered some of the issues I remember reading about trying to do physics on discrete space-times.
The first issue is that you cannot have nice, neat, orderly lines. In order to approximate the observed Lorentz invariance and local isotropy, the points have to be scattered about in a chaotic fashion.
The second issue, which I understand is considered a killing blow, is that all known theories of discrete space-times predict that light from distant sources should be blurry, but not even the slightest blurring is observed.
The first issue is that you cannot have nice, neat, orderly lines. In order to approximate the observed Lorentz invariance and local isotropy, the points have to be scattered about in a chaotic fashion.
The second issue, which I understand is considered a killing blow, is that all known theories of discrete space-times predict that light from distant sources should be blurry, but not even the slightest blurring is observed.
- #124
balkan
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ridiiiiicuuloouuuuussssss![/size]
- #125
robert Ihnot
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I asked myself similar questions a few years back. How do I know the commutative law is true etc. Anyway I hope this helps.
To my mind, if you go to the supermarket and pay for two pops and three cans of tuna, the price is not changed if you pay for the tuna first and then the pop, or for that matter if you mix the items up at check out.
Now if things were not like that, then if a primitative man put three arrow heads and two pieces of wood in a hiding place, the material could change if he did not extract them in the same order.
If during roll call, they started out by lining up from Z to A rather than from A to Z, would the number of members change?
So how could any other system possibly exist?
To my mind, if you go to the supermarket and pay for two pops and three cans of tuna, the price is not changed if you pay for the tuna first and then the pop, or for that matter if you mix the items up at check out.
Now if things were not like that, then if a primitative man put three arrow heads and two pieces of wood in a hiding place, the material could change if he did not extract them in the same order.
If during roll call, they started out by lining up from Z to A rather than from A to Z, would the number of members change?
So how could any other system possibly exist?
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- #126
Cheman
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I think a more interesting question would be to ask why does multiplication work or division?
ie - when we do multiplication, long and short, the is a set algorithm whihc we fulfil. eg - to multiply 234 by 5 we would times 4 by 5, etc. Same with long and short division.
I would be very interested to know why these algorithms actually work and whether they can be prooved.
Thanks in advance.
ie - when we do multiplication, long and short, the is a set algorithm whihc we fulfil. eg - to multiply 234 by 5 we would times 4 by 5, etc. Same with long and short division.
I would be very interested to know why these algorithms actually work and whether they can be prooved.
Thanks in advance.
- #127
matt grime
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Your asking to know why long mulitiplication works? Honestly? Or what multiplication means, ie why does it distribute. Well, for a useful introduction can i suggest
www.dpmms.acm.ac.uk/~wtg10[/URL]
there is a link to Gowers's informal writings, one explains why multipliation is commutative I think, though it is to do with sets.
www.dpmms.acm.ac.uk/~wtg10[/URL]
there is a link to Gowers's informal writings, one explains why multipliation is commutative I think, though it is to do with sets.
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- #128
FulhamFan3
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ugh. These post are stupid. Even stupider is that people take these seriously and give long drawn out explanation.
He wants proof that two quantities put together make a bigger quantity? That's called an axiom. There is no proof because is a basic obvious thing. If someone does come up with a proof I'm going to ask for a proof of the basic statements they used to prove it. It's all got to start somewhere. At some point things are just obvious.
He wants proof that two quantities put together make a bigger quantity? That's called an axiom. There is no proof because is a basic obvious thing. If someone does come up with a proof I'm going to ask for a proof of the basic statements they used to prove it. It's all got to start somewhere. At some point things are just obvious.
- #129
Cheman
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"Your asking to know why long mulitiplication works? Honestly?"
Well, yes - a proof for long multiplication and division must clearly exist, as the methods we use are simply algorithms. eg - it is obvious that 2 + 1 =3; its an axiom, but when we want to work out what 456*34 is or 6879/43 there is a specific set method we follow. This is an algorithm. I was just wondering why these algorthims work?
Thanks.
Well, yes - a proof for long multiplication and division must clearly exist, as the methods we use are simply algorithms. eg - it is obvious that 2 + 1 =3; its an axiom, but when we want to work out what 456*34 is or 6879/43 there is a specific set method we follow. This is an algorithm. I was just wondering why these algorthims work?
Thanks.
- #130
Curious3141
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Cheman said:"Your asking to know why long mulitiplication works? Honestly?"
Well, yes - a proof for long multiplication and division must clearly exist, as the methods we use are simply algorithms. eg - it is obvious that 2 + 1 =3; its an axiom, but when we want to work out what 456*34 is or 6879/43 there is a specific set method we follow. This is an algorithm. I was just wondering why these algorthims work?
Thanks.
Long multiplication ? Depends on distributivity of multiplication over added expressions :
The following is written out to correspond to the right to left (least significant to most significant) method taught in elementary school. When multiplying by a product of ten, shift one place left, by a product of hundred, two places left. Those should be obvious.
456*34 = 456*4 + 456*30
= (4*100 + 5*10 + 6)*4 + (4*100 + 5*10 + 6)*3*10
then just add it all up. The addition of remainders to the next most significant digit depends on associativity of addition.
Long division similarly depends on distribution i.e. \frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}. I'm sure you can see why.
- #131
K.J.Healey
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So what you're saying is you can't prove addition?
Just kidding.
Just kidding.
- #132
JasonRox
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Healey01 said:
Crabs is a very insightful person, who came up with such a beautiful question. :zzz:
Just kidding. I hope he/she stops smoking crack.
- #133
MattTuc13
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The proof for addition is simple...
THE DEFINITION!
We define addition by adding numbers together to find their sum, therfore, that is the proof.
Descarte-Philosopher whose most known statement is "I think therefore I am".
Before he came to this conclusion, he said that the only thing that can be proven in the world is math. Why? Because by definition it is true.
What you are doing is trying to prove something that by definition is already true.
THE DEFINITION!
We define addition by adding numbers together to find their sum, therfore, that is the proof.
Descarte-Philosopher whose most known statement is "I think therefore I am".
Before he came to this conclusion, he said that the only thing that can be proven in the world is math. Why? Because by definition it is true.
What you are doing is trying to prove something that by definition is already true.
- #134
HallsofIvy
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FulhamFan3 said:
An axiom? It's not even true. -7 "put together" with -4 (addition) is -11. That's not a "bigger quantity". all you are saying is that you do not understand the original question.
- #135
FulhamFan3
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HallsofIvy said:
math with negatives is equivalent to subtraction. the question is can you prove addition, not subtraction.
- #136
metrictensor
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this is stupid
In your proof you have assumed addtion by applying subtraction. The problem is here that you don't understand what is meant by proof. My explanation will appear below.
In your proof you have assumed addtion by applying subtraction. The problem is here that you don't understand what is meant by proof. My explanation will appear below.
Gecko said:
- #137
Hurkyl
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Actually, additive inverses are usually defined first, then subtraction defined in terms of that.
- #138
metrictensor
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First off good question dr. Don't fret if 99% of the answers you get here are obviously from people who have never thought before. Short answer to your question is 'by definition'.
The problem, as I see it, is with what is meant by proof. All those mathematical proofs are useful within mathematics but what about in the everyday world. Ask a child what 1 + 1 is and they will say 2. How did they learn that? Your question is the same as asking 'why are all unmarried men bachelors?' The only reason is by definition. If you can understand 1 you can understand any number. Ask someone how many parents they have and they will say two. The reason 1 + 1 = 2 is the same as why the letter 'b' follows 'a'.
It is not that the mathematical explanation given above is "wrong". It is only appropriate in a particular context. How do we know this? In everyday life even people who have never seen the (so called) proof understand that 1 + 1 = 2. They know this by definition.
In most cases, anyway, the problem arises from a misunderstanding of the word proof. What is even more astonsihing is how little is gained when we answer this question. I understand perfectly the answer to your question. The benefit comes not from finding an answer but in the understanding we get from seeing how we went about finding the answer.
The problem, as I see it, is with what is meant by proof. All those mathematical proofs are useful within mathematics but what about in the everyday world. Ask a child what 1 + 1 is and they will say 2. How did they learn that? Your question is the same as asking 'why are all unmarried men bachelors?' The only reason is by definition. If you can understand 1 you can understand any number. Ask someone how many parents they have and they will say two. The reason 1 + 1 = 2 is the same as why the letter 'b' follows 'a'.
It is not that the mathematical explanation given above is "wrong". It is only appropriate in a particular context. How do we know this? In everyday life even people who have never seen the (so called) proof understand that 1 + 1 = 2. They know this by definition.
In most cases, anyway, the problem arises from a misunderstanding of the word proof. What is even more astonsihing is how little is gained when we answer this question. I understand perfectly the answer to your question. The benefit comes not from finding an answer but in the understanding we get from seeing how we went about finding the answer.