Daymare17

Your country? I thought it was Donald Trump's and Frank Carlucci's and Bill Gates' country?

Do you realise that you just opened the door to insanity? If it does not have to be consistent with the real world, then anything goes! All scientists are aliens. Einstein was a scientist. Therefore, Einstein was an alien! Practice is the touchstone of every theory. I don't really understand what you mean. Can you put it another way? The same instant? Pardon me sir, but how long is an instant exactly? Is it an infinitesimal interval of time? But then the apple will be subject to minute changes. Or is it no time at all? A zero of time - what a meaningless fancy. Everything exists in time; time is consequently a fundamental element of existence. The axiom x = x signifies that an object is equal to itself if it does not change, that is, if it does not exist. That's right - an instant does not exist. It is just another figment of your mind. I shall commend you on your very lively imagination. But imagination only makes good science fiction, not good science. What we have here is a prime example of philosophical idealism - you take the abstraction, which is after all only an approximation of the real world (i hope you don't disagree with that), and hand it back to the real world, expecting the real world to follow your dictat. It is a terrible shame that so many scientists and science-interested people are ignorant or contemptuous of philosophy. So many - there is no milder expression - so many infantile mistakes could be avoided!

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.

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.

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. 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. And who's Doron Shadmi anyway?

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'm sure you can. But please give me an empirically verifiable example. If the criterion is not empirical verifiability then anything goes. 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. Yep. And if it's not applicable then we should try to make it so, no? 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.

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.