Is Addition Really a Basic Skill?

[NeutronStar]

Now, you could have just said that irrational numbers don't satisfy whatever concept and left it at that...

Ok, irrational numbers don't satisfy the condition of being individual things because they aren't individual things.

All irrational numbers are a result of self-referencing situations. Thus they are relative concepts referencing relationships between interconnected (self-referenced) concepts rather than being individual quantities in their own right.

They arise from self-referenced reflections. Like looking in a mirror. They are illusions. They don't exist outside of their self-referenced situations.

Does that make it clear?

Probably not.

Actually the deal with the irrational numbers is really just an aside to the bigger picture. This is only one of many enlightening concepts that fall out of correcting the definition of the number One as it is based on set theory.

 

[jcsd]

Sigh I wasn't going to argue, but Neutron Star you really do need to realiz ethat you much of these disagreemnts are the result of your own misconceptions and not only that, but your not going to persuade anyone otherwise because this fact is obvious to all!

George Cantor considered a representation of real numbers to prove a general property of the real numbers, nowt wrong with that. But you are cofusing the represntation with the actual mathematical object. Where exactly do we need to round THIS represnation of THIS irrational number: $\sqrt{2}$? (and note that this representation is a much more useful one than 1.41421...)

 

[NeutronStar]

Sigh I wasn't going to argue, but Neutron Star you really do need to realiz ethat you much of these disagreemnts are the result of your own misconceptions and not only that, but your not going to persuade anyone otherwise because this fact is obvious to all!

Well, just for the record, I didn't start a thread of my own to convince anyone of anything.

I just keep coming back to this thread and responding to questions because I have no real discipline. (ha ha)

But you're right this really is pretty fruitless on my part. Besides, even if I actually got through to someone what good would it do me? They would probably just run off and take credit for the idea anyhow and no one would ever know who NeutronStar even was!

There's really no point in wasting my time here further. In fact, I retract everything I said. I was grossly wrong about everything because I'm a stupid uneducated moron. I see that now. Thanks for pointing it out.

 

[jcsd]

I thought you would retract evrything you said! :)

But seriuosly my point is that I am understanding everything you are saying, but I also undretsand why what you are saying is wrong, infact many of your observations are not even new, we're going over some very well-tordden territory here.

 

[Hurkyl]

Who's doing this ?,...

I'm pretty sure you mentioned earlier about truncating decimal representations of irrational numbers, and you did again in this post, so I hope you can understand why I get the impression that nonrepeating decimal expansions are a significant part of your problems with irrational numbers.


Cantor doesn't assume that real numbers are decimal expansions: just that there is a bijection between real numbers and decimal expansions.


Cantor's argument can even be written purely analytically:

Suppose f is any function from the positive integers to the real numbers.

Then, compute the following:

$$\phi := \sum_{n=1}^{\infty} 10^{-n} \left( 9 - \lfloor 10^n f(n) \rfloor + 10 \lfloor 10^{n-1} f(n)} \rfloor \right)$$

Then you can show that $f(n) \neq \phi$ for any positive integer n.

 

[NeutronStar]

Cantor doesn't assume that real numbers are decimal expansions: just that there is a bijection between real numbers and decimal expansions.

Well that's an assumption that I'm not prepared to make. How does Cantor justify such an assumption?

This goes back to the dog name "43". Just because we can create a string of numerals doesn't mean that we have created a valid and meaningful concept of number. That's simply invalid logic.

This is especially true when those strings are endless decimal expansions. Cantor's assumption that there exists a bijection between the reals and all possible decimal expansions doesn't hold water for me.

Look at it this way.

All irrational numbers can be represented by non-repeating decimal expansions.

That's true!

But to then conclude that all non-repeating decimal expansions represent irrational numbers is incorrect logic! It's simply wrong. Any logic instructor will quickly tell a first-year college student that such a backward conclusion is illogical and doesn't hold water.

Here's another example.

I claim that all irrational numbers are a result of a self-referenced situations.

While I believe that this is true the opposite is not true.

In other words, all self-referenced situations do not give rise to irrational numbers!

Cantor is assuming (with very incorrect logic) that all non-repeating decimal expansions qualify as irrational numbers. But he has absolutely no logical basis to make that claim. It simply isn't logically sound reasoning. Therefore I deny the bijection assumption upon which his summation equation rests.

Sorry. I'm not trying to be difficult. I'm just being logical.

 

[arildno]

Cantor's diagonal argument constructs an increasing, bounded sequence of rationals whose limit cannot be on his list.

But from what I know, we partition the set of ALL such sequences into equivalence classes, which are then identified with the reals.
Hence, what Cantor constructs, is a REAL number.

I might be wrong on this, and deserve a scathing from several sides..

 

[matt grime]

So far we have ascertained that the only collections of objects you will allow to be called sets are collections of (possibly infinitely many) natural numbers.

This of course begs the question of why you accept the notion of a bijection from N to R since R isn't a set, and functions are defined for sets (usually).

The Real numbers are equivalent to the set of cauchy equivlance classes of convergent strings of rational numbers, or the dedekind cuts (are rationals acceptable, why?), and thus given any element , x, of this complete metric space, it is easy to construct a decimal expansion, which modulo recurring 9s, which we will formally declare not to happen, ie replace them with 0s instead. It is then rather easy to construct bijections between R and and P(N) the power set of the natural numbers. Is the power set operation allowed in your theory? Ie is P(N) a set? Anyway, given P(N) we can apply the theorem that no set is bijective with its power set (a result that is true for all sets in the model of ZF wherein we are operating). So what's wrong here?


What on Earth is a self referenced situation?


You also appear to be arguing that the only sets with quantity are finite ones. I doubt that many here would disagree, per se, and that all other sets are infinite and that there is no quantitative difference between them. Well, that is then up to you to state what you mean by quantitative for infinite sets.

I agree that A implies B does not imply B implies A, your irrational/repeating decimal bit, but since we have a theorem that states all rationals have an eventually repeating decimal expansion (I allow of 0 recurring here rather than saying terminating), actually A iff B holds.

Note, though, that at no point in Cantor's argument does it say that any of the numbers on the list are or aren't rational, nor that the decimal so constructed is or isn't rational. Merely that given any countable list of real numbers there is a real number not on that list.

And, since a decimal represents a series, which has convergent partial sums that form a cauchy sequence of rationals, then it is a real number.

I'm sorry that you don't accept that, but your acceptance (or even understanding) is not required for it.

 

[Hurkyl]

Actually, my analytic version of Cantor's argument is slightly off -- I tried a bad shortcut. Here's a valid version:

Let g be the function on the set {0, 1, 2, ..., 9} where g(0) = 1 and g(n) = 0 for n > 0.

Let f be any function from the positive integers to the reals. Compute

$$\phi_f := \sum_{n=1}^{\infty} 10^{-n} g(\lfloor 10^n f(n) \rfloor - 10 \lfloor 10^{n-1} f(n) \rfloor )$$

Then $\phi_f \neq f(n)$ for all positive integers n.

 

[NeutronStar]

On Self-reference and Irrational Numbers

What on Earth is a self referenced situation?

Well let's see,…

Pi is the ratio of two properties of the same geometric object. Changing one of those properties coincidentally changes the other. In other words, you can't adjust the diameter of a circle without directly affecting the length of its circumference and vice versa.

The irrational number Pi arises from a self-referenced situation much like putting two mirrors back to back. Trying to adjust the image in one mirror has an instantaneous affect on the image in the other mirror.

The $$\sqrt2$$ is the number than when collected together the same number of times that it represents adds up to 2.

Think of it in terms of collections of things. Let $$q=\sqrt2$$ then q is the number that must be collected together the same number of times that it represents. In other words if we let q=2 then we must collect 2 together twice. But that equals 4 so that won't work. So let q=1. Then we must collect 1 together once. But that only equals 1 so that won't work.

So q must be somewhere between 1 and 2. Let's try 1.5 Well, that means that we have to collect 1.5 together 1.5 times. So collecting it together once gives us 1.5. Then collecting .5 more of it we get 2.25. Well, we're getting closer but we're still not right on the money, we're a little bit over.

As we continue this process we find that if we let q=1.4 then collecting that together 1.4 times gives us 1.96. Hey! That's pretty darn close to 2. In fact, if we could collect 1.4 together exactly 10/7 times we'd have it exactly! But wait a minute!

We can't do that because it's a self-referenced situation! Remember that q represents BOTH the number that we are collecting together AND the number of times that we must collect it. So q can't be both 1.4 and 10/7 simultaneously. If we change the number of times we collect q together, then we have also changed the value of the quantity that we are collecting.

So we're stuck in this self-referenced situation just like the back-to-back mirrors. Adjust one mirror and it instantly changes the image in the other mirror. The square root of 2 is a self-referenced situation. In fact, the square root of all numbers are self-referenced situations. Obviously all self-referenced situations don't result in irrational numbers. The square root of 4 for example is simply 2.

Let's try one more,…

Euler's number $$e$$ is defined as $$e=\lim_{n\rightarrow \infty} \left(1+\frac{1}{n}\right)^n$$

Here we can see that n is being self-referenced as it is in an expression that is being taken to the power of n itself. There are other ways to define $$e$$ but I think that this definition clearly shows the self-referencing situation that's involved.

I have asked many mathematicians over the course of my life to present me with a "meaningful" irrational number that is not associated with a self-referenced situation. So far no one has been able to come up with an example.

The closest thing that they can offer are arbitrary strings of meaningless numerals. When they give me such meaningless strings of numerals I feed them to my dog named "43".

Although, I have yet to even see an arbitrary string of non-repeating numerals that isn't self-referenced in some fashion. Take the meaningless so-called irrational number 1.12123123423451234561234567 and so on. Just for clarity that's just 1, then .12, then 123, then 1234 and so on to create a non-repeating endless decimal. Then using Cantor's backward logic and claim that this string of numerals represents a valid concept of number.

Ok, just for fun let's say that we accept this as a valid number. Then I still claim that it is self-referenced because as the sequence grows it's dependent upon what came before it. In other words, each new additional 123,.. must be different from the previous 123,…. otherwise we'd have no way of proving that it never repeats. So self-reference is crucial to its very definition of being non-repeating. Therefore the property of self-reference is mandatory in this situation.

However, even if someone could give me a string of numerals that does not repeat and is not self-referenced I would still be skeptical that it satisfies the definition of a number. It just looks like dog food to me.

What I would really like to see is a "meaningful" irrational number (like pi or e) that is clearly not the result of a self-referenced situation. Then I could know that I'm wrong and retract my conjecture that all irrational numbers are the result of a self-referenced situation. Thus far in my life I have yet to see any such example.

 

[matt grime]

So you're going to ignore the maths and simlpy say that "because I can only think of one way of describing it, then that's it"? Well, as sqrt(2) isn't the number which when gathered together "itself" number of times we get 2, then we're ok, I suppose.

Why on Earth do you not accept decimal expansions? Ok, they aren't perfect (diadic representations), but they're useful.

Of course, and I'm willing to stake my house on this, you don't actaully know what the real numbers are do you? I don't mean that in any "gee, look what the stupid kid thinks" way, but as a common observation that, as we don't actually live in a platonic realm, then the real numbers need some rigorous foundation, one that isn't taught to the vast majority of people. One of which is as infinite strings of decimals.

I can give a non-self referential proof, perhaps, of sqrt(2), but as your idea of self referential is hand-wavy and vague, who knows.

Let's try anyway. Sqrt(2) is the dedekind cut given by in informal presentation, sup{x in Q : x^2<2}, or by the cauchy sequence of finite sums given by x_n is the maximal element in the set of rationals with no more than n non-zero terms whose square is less than 2.

Just because you interpret multiplication in that way of yours neither means that that is what it is, nor that even doing so is the only way.

The real numbers are "the" complete totally ordered field.

Of course, since you think "gathered together sqrt(2) many times" is how we define sqrt(2), then we could be in for an uphill battle. Moreover, your argument presumes that sqrt(2) exists and must in some way represent some quantity of which we can say: gathe something together sqrt(2) times, where as really it is a construction derived from the rationals, themselves derived from the integers, in turn derived from the naturals, which we'll presume exist, though they can be put on a rigorous footing if need be.

Also, e isn't usually defined to be the limit you give, and even if it were, then all rational (non-integral) numbers are defined in a self referential way: 1/n is the number that when gathered together n times gives 1, is a self referential formula, by which you appear to mean "has some symbol appearing twice", that defines 1/n. Oh no, better reject rationals. Actually, you really had better reject them, hadn't you... not to mention integers: surely k is the thing which when split k ways equally gives 1 to each, k referring to itself... but wait, what's 1?

I particularl like your observation that there are other ways to define e, but you don't list them, instead you only give one that supports your claim, when the claim surely is "for all", and proof by example will never do there.


e is sum over n in N of 1/n!

is that self referential since n appears twice even though it isn't a variable (it is a dummy variable, and index, is that ok?)?

Ok, what about e is the unique value of f(1) where f is the unique solution to f'=f with boundary condition f(0)=1?

Your particularly bizarre argument involving 10/7 and 1.4 is very strange indeed. Who said that collecting 1.4 10/7 times ought to imply that they are equal and that they are the sqaure root of 2? Apart from you I mean.

 

[NeutronStar]

Your particularly bizarre argument involving 10/7 and 1.4 is very strange indeed. Who said that collecting 1.4 10/7 times ought to imply that they are equal and that they are the sqaure root of 2? Apart from you I mean.

We obviously have some miscommunication here because I never even implied that collecting 1.4 10/7 times ought to imply that they are equal and that they are the square root of 2. In fact, I thought that I clearly stated that they can't be the square root of two because q can't be two different numbers simultaneously! That was my whole point. The number q represents a single number that must represent BOTH the number we are collecting together, and the number of times that we are collecting it. My whole point is that it's a self-referenced situation.

Your assertion that I believe that collecting 1.4 together 10/7 times ought to imply that these numbers are equal and that they are the square root of 2 is simply not what I said.

As far as I can see the whole idea of collections and their behaviors was the initial concept that sparked the whole idea of set theory. If the theory has be distorted so terribly that it can't even support this fundamental basic concept then that's really scary! Things are worse than I thought!

 

[jcsd]

The thing is though sqrt(2) clearly is a real number by the defintion of a real number (there are sveral equivalent defintions, but however which way you choose to play it that sqrt(2) is a realmnumber is unescapable!). Your working from some vague preconceived notion of a real number that you have inside your own head, but mathematicians already have a rigorous definition of a real number and if your preconceived notion does not fit this definition then you are not talking about rela numbers in the commonly understood sense.

 

[learningphysics]

Definition of addition and proof of basic laws

Hi Drcrabs. I saw your question, and I think I understood what you were looking for. Have a look at this site:

http://ndp.jct.ac.il/tutorials/Discrete/node50.html

They first define addition, and then prove the commutative and associative laws of addition (basic laws that we use day to day). It involves a lot of proof by induction.

I asked myself similar questions a few years back. How do I know the commutative law is true etc. Anyway I hope this helps.

 

[Hurkyl]

I'm sure you're familiar with the construction of the complex numbers from the reals, right? You can consider the complexes as ordered pairs (x, y) (i.e. the complex plane), and define addition and multiplication for them, etc.


You can do an entirely analogous thing for the square root of 2.


I'm going to define a number field (I don't know why it has that name, but it does) that is often named $\mathbb{Q}[\sqrt{2}]$.

Take all ordered pairs of the form (p, q) where p and q are rational numbers. Here are the definitions for arithmetic operations:

(a, b) + (c, d) = (a + c, b + d)
(a, b) - (c, d) = (a - c, b - d)
(a, b) * (c, d) = (ac + 2bd, ad + bc)
$$\frac{(a, b)}{(c, d)} = \left( \frac{ac - 2bd}{c^2-2d^2}, \frac{bc - ad}{c^2 - 2d^2} \right)$$

I can define an ordering relation, <, but it's a little more complicated requiring several cases.


We have the embedding of the rationals in this number field via:

a --> (a, 0)

(just like for complex numbers)

And it's fairly straightforward to check that:

(0, 1) * (0, 1) = (2, 0) = 2.

So, (0, 1) is a square root of 2, with the other being (0, -1).


And, just like we usually write the complex number (a, b) as a + bi, we would usually write the element (a, b) of this number field as a + b &radic;2

 

[drcrabs]

Hi Drcrabs. I saw your question, and I think I understood what you were looking for. Have a look at this site:

http://ndp.jct.ac.il/tutorials/Discrete/node50.html

They first define addition, and then prove the commutative and associative laws of addition (basic laws that we use day to day). It involves a lot of proof by induction.

I asked myself similar questions a few years back. How do I know the commutative law is true etc. Anyway I hope this helps.

Cheers bro

 

[matt grime]

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.

 

[Hurkyl]

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.

 

[balkan]

ridiiiiicuuloouuuuussssss!

 

[robert Ihnot]

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?

 

[Cheman]

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.

 

[matt grime]

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.

 

[FulhamFan3]

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.

 

[Cheman]

"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.

 

[Curious3141]

"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.

 

[K.J.Healey]

So what you're saying is you can't prove addition?
Just kidding.

 

[JasonRox]

So what you're saying is you can't prove addition?
Just kidding.

Crabs is a very insightful person, who came up with such a beautiful question. :zzz:
Just kidding. I hope he/she stops smoking crack.

 

[MattTuc13]

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.

 

[HallsofIvy]

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.

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.

 

[FulhamFan3]

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.

math with negatives is equivalent to subtraction. the question is can you prove addition, not subtraction.

 

[metrictensor]

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.

1+1 = 2
subtract one from both sides
1 = 1

lol.

 

[Hurkyl]

math with negatives is equivalent to subtraction. the question is can you prove addition, not subtraction.

Actually, additive inverses are usually defined first, then subtraction defined in terms of that.

 

[metrictensor]

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.