# x not= x

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A mathematical proof is counted as valid if it is consistently verifiable using accepted axioms. But how about these axioms? Let us take the most fundamental and universally accepted axiom (in fact, it's so universally accepted that it hasn't even been officially listed as one). Let's take the axiom 1 = 1, or x = x.

In first grade books, integers are depicted as fruit. An apple plus an apple equals two apples. This is a useful concretisation. If mathematics is correct, then this kind of concretised example must be correct, since "the proof of the pudding is in the eating" - the proof of the theory is in its applicability to the real world.

There are serious problems with this apparently unquestionable axiom. First of all, it presupposes that there exist identical things. The very act of "adding" the two apples together requires that the apples are identical. Because after you add them together, you do not have one small apple plus one slightly larger apple, as you would in reality: you have "two apples". By your act of putting them in the same basket they have now, according to mathematics, been fused into an indistinguishable mass, the "sum". Cut the sum in half, and two identical apples emerge.

Some might object that the axiom does not state this, it simply states that the apple is equal to itself. To be sure, this is a useful approximation. The apple certainly is not equal to very much else. Nonetheless, it is incorrect and every mathematical formula is therefore also incorrect. I shall attempt to prove here that the apple is not equal to itself, that 1 not= 1.

Everything changes continously. The essence of matter is movement. Einstein explained this: Mass and movement (energy) are inseparable; in fact, the same thing. Thus, the apple is not equal to itself since it is changing all the time. X not= x. 1 not= 1.

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This is why mathematics is historically one of the liberal arts.

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And also why many degree courses give a BA instead of a BSc.

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Damn, I've been doing my mathematics with numbers and not apples. Wonder if they'll revoke my BA? But I guess that's the problem when they don't teach degree level maths from first grade books.

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Apple = Apple - change. is that what youre trying to say then?

more like Apple <--->Apple (change acceptable)

the Equals sign (=) does not factor in Change where apple A is a constant.

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I think you should separate abstraction and application. You can perfectly create the natural numbers from a simple axiom using set theoretical arguments. Now when you would like to apply this system you decide that the unity is an apple regardless of its size, colour, configuration, weight etc...... If you disagree with this decision and find that ambiguous and consider that a red apple is not the same as a green one, then that in no way changes the abstract system. The only comment you are making is on the application of the abstract system to the real world.

Mandrake

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apples or similar fruit is just an abstract way to explain a young child what 1+1 means and why it equals.. mmmmm.... damn where are my bananas

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apples or similar fruit is just an abstract way to explain a young child what 1+1 means and why it equals.. mmmmm.... damn where are my bananas

I think you're standing it on its head. Numbers are much more abstract than bananas. Fruit is a concrete way to explain it. My point is that if an abstract system doesn't fit with reality then there's something wrong with the system, plain and simple. 1 + 1 doesn't really "mean" anything at all, apart from the bananas. Of course, 1 + 1 is a gigantic conquest of the human mind. We'd be worse off without it. But we should remember that it's unperfect.

I think you should separate abstraction and application. You can perfectly create the natural numbers from a simple axiom using set theoretical arguments.

I'm sure you can. But please give me an empirically verifiable example. If the criterion is not empirical verifiability then anything goes.

Now when you would like to apply this system you decide that the unity is an apple regardless of its size, colour, configuration, weight etc...... If you disagree with this decision and find that ambiguous and consider that a red apple is not the same as a green one, then that in no way changes the abstract system.

Of course. But it does point out that the abstract system is erroneous and should be changed to accomodate for the discovery. How, I'm not so sure.

The only comment you are making is on the application of the abstract system to the real world.

Yep. And if it's not applicable then we should try to make it so, no?

Apple = Apple - change. is that what youre trying to say then?

more like Apple <--->Apple (change acceptable)

the Equals sign (=) does not factor in Change where apple A is a constant.

Yes. One, or apple, is an abstraction. It's like the net that you use to catch the real world. But the real world is always changing and, for it to be precise, your net should change too, "tightening out" or "loosening up" as necessary. But 1 doesn't. It's just 1, eternal 1. It's a totally false and static abstraction. It simply doesn't square with reality.

And flowing from this is my argument that mathematics is based on a fundamental fallacy and should be revised somehow. Can anyone who's smarter and less lazy than me try to explain to some degree what implications the axiom X not= X has for the rest of mathematics? I'd try it myself but I'm too tired right now.

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Look, Daymare, mathematics *is* an abstract subject. If reality does diverge from a model then the model is wrong not the underlying mathematics that are independent of the use in the model. Deal with it. Stop trying to use unmathematical arguments to cast shadows on mathematics itself. 1+1=2 is, frequently, the definition. There is no fallacy in the mathematics, even if there were a fallacy in the model which originally motivated it. What on earth in reality is 1? Platonism is not very popular in mathematics. So get a new model if you're that annoyed with it, but leave mathematics alone, please.

If you are going to introduce the axiom that x =notx, then you'd better state in what axiomatic system you're operating. I mean, is x a group element, a vector space, a set, a proposition??? It's just meaningless nonsense.

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There is nothing reall I can say that hasnt been said before. a number cannot be warped through time...it cannot have greater mass than another...its an abstract. Mathmatics as a whole is abstract! Thus, adding two apples is simply a "real world" physical intperpretation of the equation. I have two spheres (this is an abstract) one sphere is exactly like the other. Thus, x = x...now if I were to be critical, one sphere is less dense than the other...and thus x =/= y (y being the less dense sphere) Since no equation uses (x) for two different numbers (x=2 and x=7)...that would be a bit confusing, so we use x,y,z,a,b,c...ect. So if two apples are different, they would not both be represented by the same variable. and as we all know, x does not always = y

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You see, this is why maths should be taught properly at school. We are not saying the apples are equal, or not eqaul. We are saying things about the NUMBER of apples, not the apples themselves.

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You see, this is why maths should be taught properly at school. We are not saying the apples are equal, or not eqaul. We are saying things about the NUMBER of apples, not the apples themselves.

I knew there was something about why all this about xnot=x didn't fit...I think you got it there, yes.

I'd like to see someone say that you cannot claim that 2 apples are never two apples, no matter which appels you've got.

Another example. Say you have 3 chairs : one arem chir, one deckchair and one stool. Youc an say swap the stool for a dinning chair. The stool certainly is different from the dinning chair, but you still have 3 chairs.

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If there WAS a flaws in mathematics that u suggest , then i dont think the great thinkers like Plato, Archimides, Pythogoras etc , or great mathematicians like Gauss, Euler, Cauchy, Newton etc. would have overlooked it so easily. but the "paradox" has already been solved by the famous Matt Grime. Remember you saw it here first. at ScienceForums.

I will start selling "I talked to Matt Grime" t shirts soon.

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You should also have the "I got a reply from Matt when he'd not had any coffee yet that morning and was a shirty little sod".

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I knew there was something about why all this about xnot=x didn't fit...I think you got it there' date=' yes.

I'd like to see someone say that you cannot claim that 2 apples are never two apples, no matter which appels you've got.

Another example. Say you have 3 chairs : one arem chir, one deckchair and one stool. Youc an say swap the stool for a dinning chair. The stool certainly is different from the dinning chair, but you still have 3 chairs.[/quote']

I think you are missing the whole point of abstraction here. The whole point of applying the abstract consistent number system is that you can quantify something. Depending on what is your question the number system can help you quantify. If you ask for three apples, then someone can give you red or green ones. If you specify that you want red apples,...well you get the point. But that doenst have any impact on the number system. If you consider that a red apple is not the same as a green one, then you cant call it both "x", i.e., it is not the same variable.

The whole point of talking in terms of apples/parts of a pie or whatever is to illustrate abstract mathematical concepts and not define them !

Since it is a little hard to start in kindergarten with real mathematics, it is done more intuitively.

Mandrake

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You should also have the "I got a reply from Matt when he'd not had any coffee yet that morning and was a shirty little sod".

That's on the back.

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"I will start selling "I talked to Matt Grime" t shirts soon."

Can I have the black XL version of "Matt Grime - Kicking math-haters in the arse since the dawn of time" ?

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dude, an apple is not defined by size. one being smaller than another makes it no less of an apple than the one it is compared to. so, if you have an apple and are given another, you have two apples. it isn't a hard concept.

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I'd rather have "matt grime 1 - doron shadmi {}"

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I'd rather have "matt grime 1 - doron shadmi {}"

Haha this one is funny. Though i think doron would say 1_1 or 1_0_1

Mandrake

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It seems all the talk about apples was a little confusing, judging from the replies of MandrakeRoot and matt_grime. But just look at what i said:

"In first grade books, integers are depicted as fruit. An apple plus an apple equals two apples. This is a useful concretisation. If mathematics is correct, then this kind of concretised example must be correct, since "the proof of the pudding is in the eating" - the proof of the theory is in its applicability to the real world."

My point is that an apple is never equal to itself, nor is anything else for that matter. It's just a mass of particles that keeps changing. That's what I mean when I say that 1 = 1 is uncorrect.

How silly is it to claim that maths does not have to fit in with the real world. It's basically the claim of the Pythagoreans, who thought that everything was Number. It's like saying that everything is ideas, or for that matter, God. The world exists apart from our ideas, and we have to struggle to keep our ideas in tune with the world. If an idea is old, unprecise and incorrect it should be cleaned out somehow.

Look, Daymare, mathematics *is* an abstract subject. If reality does diverge from a model then the model is wrong not the underlying mathematics that are independent of the use in the model. Deal with it.

First, you say that mathematics is abstract. And then you imply that mathematics is not a model! Every abstraction is a model.

The subject of my post was x = x. That proposition is what I said was untrue. Then you begin talking about models and how models are independent of mathematics. Where did you get this idea about 'models' from? How is x = x not a fundamental proposition of mathematics? How is it independent of mathematics? If x = x is incorrect (= unprecise), then it means that the whole body of maths is incorrect.

It seems that you think mathematics has a "blank cheque" in relation to the real world. This kind of fetishism is harmful. One must know how adapt schemas to facts.

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I think you're standing it on its head. Numbers are much more abstract than bananas. Fruit is a concrete way to explain it.

hmmm... ok i was wrong. however' date=' for a young child it may be the most abstract comparison, since he does not have the mental capability yet to go any further in abstraction.

My point is that if an abstract system doesn't fit with reality then there's something wrong with the system, plain and simple.

reality is something that lies in the view of the beholder. so the view of a child is very basic and for his reality it's ok to say every apple counts as 1. when you make the definition "1 object = 1" then it's regardless of any size or other difference anyway. this does not depend on who is looking at it. so if you would make the definition "1 small object = 1" and "1 big object = 2" then u still can say 1 == 1. so my point is, that it depends on how you interprete reality and what rules you define.

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And I didn't just say that one apple is different from another apple. I think you guys misunderstand me. What I said is that any apple is never equal to itself.

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reality is something that lies in the view of the beholder. so the view of a child is very basic and for his reality it's ok to say every apple counts as 1. when you make the definition "1 object = 1" then it's regardless of any size or other difference anyway. this does not depend on who is looking at it. so if you would make the definition "1 small object = 1" and "1 big object = 2" then u still can say 1 == 1. so my point is, that it depends on how you interprete reality and what rules you define.

Not at all. So every person has his own 'reality'? Then reality entered the world along with yourself, and before humans walked this earth, then there was no reality. Reality is objective. However our interpretation of reality is different from person to person and time to time. The objective fact is that everything is always changing, and 'x' becomes old and worn out and incorrect the very moment you write it down.

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And I didn't just say that one apple is different from another apple. I think you guys misunderstand me. What I said is that any apple is never equal to itself.

this does also depend on what you define as equal... if you look at it from the viewpoint of just the existance of an object than the object is always equal to itself. unless it rots or whatever.. but thats something different.

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