x not= x

[fairychild]

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.

i disagree.. imo every person has his own reality. his own perception of things. no brain works exactly the same as any other. if you take mind altering substances like alcohol e.g. then your reality is different. if you take LSD then your reality and perception etc also changes. if you stop breathing you get dizzy .. see what i mean?

[YT2095]

Apple needs to be a Set containing all qualifying atributes of an apple.

the term `=` could never really be used accurately, "Equivalent" would be better syntax, or "Contains properties Of..."

then Apple(parameter #1)=parameter #1 in (Apple set) could be true.

 

ie/ Apple (green) = (Green) qualifier for Apple in it`s set

it doesn`t mean it`s an apple, just that within the set of Apple, green is a match or `=`

 

 

to use Equals, you need to nail it down to Specifics, and since "an Apple" is a non specific or Variable, it`s a bad term to use.

[matt grime]

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.

 

and the point is that 1+1 is a statement about units in rings, or elements in the peano system. You're confusing reality and mathematics

 

 

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.

 

And how uninformed that you think that only those things that are real are mathematics. What is "real" anyway?

 

Equality, addition, and so on are perfectly precise and the fact that there is only one unique instantation, physically and temporally of physical objects is neither here nor there.

 

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

 

Evidently you do not know what an axiomatic system is, and a model of such, as is distinct from an application and model.

 

Whatever the original motivation of numbers is (ie it counts the cardinality of sets [of apples]) has nothing to do with describing each apple uniquely. Did you not read, or just not understand, the simple explanation that when we say 1 apple plus one apple gives two apples we are talking about a property of collections of apples (how many there are, which is reasonably well understood, as apples are reasonably discrete objects) not about the apples themselves being in any sense "the same"? That is not what the equals sign means. "1" "exists" in the loose mathematical sense, independently of your use of it to say how many apples there are, since it is nothing to do with apples.

 

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.

 

I was trying to inform you of some of the things mathematics "is", since you evidently are ignorant of the subject.

 

The idea that x=x is fundamental needs some explaining on your part. I'm happy with it - it's encoded in the ZF axioms after all, and I know what it means. You seemingly don't if you think 1 apple = 1 apple is both a mathematically sound statement (you're misinterpreting the "=" sign) and simultaneously about the apples being the same apple.

 

 

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.

 

You are confusing maths with its applications again. the only rule of thumb for a mathematical object is that it is not internally inconsistent, and is preferably non-vacuous. What exactly falls under the purview of mathematics is an endless topic of debate. If the maths we do has a use that models (and this is used in the sense you think it is, and not the sense above for axiomatics systems) things in the real world then so be it, especially if it is verifiable. The additive integers are a good way to count discrete well defined objects like apples. If I have two drops of water and I combine them how many drops do I have? When does it stop being a drop and become, say a puddle? That is not a problem of mathematics in and of itself but of your application of it to the real world.

 

In short, you are misusing the equality symbol in a most depressing way. No one says that it means the two apples are the same. In fact that is one of the first things we beat out of students - misapplications of equals signs, until such time as they are sufficiently well read to start abusing them again.

[matt grime]

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.

 

so? what's that got to do with 1=1? this is a statement about a mathematical object, yours is about whatever philosophical problems you have with adopting a non-mathematical meaning for = for apples.

 

The fact is that your interpretation for equals in the real world isn't anything to do with the mathematical one (equality is based upon, say the idea that S=T as sets iff x in S and x in T are equivalent propositions, or that [abstract] things are the same - have the same mathematical properties exactly).

[MandrakeRoot]

"In first grade books' date=' 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."

[/quote']

 

No : Mathematical statements are logically derived from the axiomatic system they are statements in (ZFK if you like) and are therefore independant from any "real-world" notion whatever that may mean.

 

The saying "the proof of the pudding..." is really the most ridiculous thing i have ever heard !

Your statement is something along the following line : "I roll a dice and roll 6, so i conclude every dice roll will always result in a 6."

 

Even if an apple changes over time, in what way does that make the apple different from himself ? If you look at two different instances of time, yes the apple has evolved and will no longer be in the same state, but when looking at the same time instance it will be the same thing.

 

Mandrake

[Daymare17]

No : Mathematical statements are logically derived from the axiomatic system they are statements in (ZFK if you like) and are therefore independant from any "real-world" notion whatever that may mean.

 

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.

 

The saying "the proof of the pudding..." is really the most ridiculous thing i have ever heard !

Your statement is something along the following line : "I roll a dice and roll 6, so i conclude every dice roll will always result in a 6."

 

I don't really understand what you mean. Can you put it another way?

 

Even if an apple changes over time, in what way does that make the apple different from himself ? If you look at two different instances of time, yes the apple has evolved and will no longer be in the same state, but when looking at the same time instance it will be the same thing.

 

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!

[MandrakeRoot]

Do you realise that you just opened the door to insanity? If it does not have to be consistent with the real world' date=' then anything goes! All scientists are aliens. Einstein was a scientist. Therefore, Einstein was an alien!

[/quote']

See you seem to get basic logic after all !

This is a logically correct statement with little pratical use.

 

Practice is the touchstone of every theory.

No it isnt. When einstein wrote down his theory most of the consequences could not be checked. It was not until much later that some empirical data could be obtained to demonstrate the "applicability" of his theory. Basically you are saying that if we are a primitive culture and cant check let's say Newtons laws' date=' newtons laws are not a proper theory.

 

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!

Yes indeed an infinitisemal instant of time. Many physical formula are derived using this notion, and like you seem to say "if it works it is true", since these formula seem to be supported by empirical data, the abstraction of using "infinitisemal instants" is a not too absurd one.

 

Mandrake

[Daymare17]

No it isnt. When einstein wrote down his theory most of the consequences could not be checked. It was not until much later that some empirical data could be obtained to demonstrate the "applicability" of his theory. Basically you are saying that if we are a primitive culture and cant check let's say Newtons laws, newtons laws are not a proper theory.

 

How did you manage to derive this nonsense from what I said? What I mean is: If the facts clash with Newton's theories, then that proves that Newton's theories are wrong. I never said anything about primitive cultures or whatever gibberish you are making up. A while ago they found out that the observed facts clashed with Newton's theories. Then they tried to mend up the theory, and it looked worse and worse after a while. Laplace (i think) took Newton to the absurd extremes. Science entered a process of searching for a new theory to fit with the facts. The process was ended with a scientific revolution, and Einstein replaced Newton. The succession of astronomical views is a good example of how theories change to fit the observed facts. Aristotle - Ptolemaios - Copernicus - etc., etc.. The theory of astronomy changed, not on a whim, but to fit the facts of the real world. The same thing should apply to mathematics. If you math people think you are some kind of special caste who are exempt from having to show proof for your theories, preferring conservative vegetation in warm corners, living off the accumulated errors of the past, then fine. Think whatever you want, but don't expect to take the science of mathematics much further.

 

Mathematics is just a way of describing the real world. Whoever mentioned anything to the effect that math is not dependent on reality - psh. Pythagorean number worshippers. All I'm asking is that math train itself for its real mission, i.e., explaining, or helping to explain, the nature of the universe. I'm not saying that abstract maths should be banned. I'm saying that every mathematician should realise the fundamentally unprecise nature of the present mathematics. X not= X, because X, whatever it is, is always changing.

 

Yes indeed an infinitisemal instant of time. Many physical formula are derived using this notion, and like you seem to say "if it works it is true", since these formula seem to be supported by empirical data, the abstraction of using "infinitisemal instants" is a not too absurd one.

 

And I quote myself: "Is it an infinitesimal interval of time? But then the apple will be subject to minute changes."

[Daymare17]

Looking over this topic, I found something amusing.

 

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.

 

Judging from what I see, if mathematics is what you pass off as "mathematics", then Platonism is not just popular in mathematics - it IS mathematics! Plato was a pure idealist. If you had a circular plate, for instance, according to him it was not a plate, it was just an unperfect expression of the Ideal Circle. This stands reality completely on its head. The abstract circle comes from the concrete round thing, not vice versa. And this error is exactly what you are making. You take mathematics, which ultimately derives from observation of objects in the real world and their relations with each other, and then you counterpose it to the real world! You say that maths has nothing to do with the real world. You separate the ideological leaves from the material trunk. One can say that you are sawing off the branch you are sitting on.

[MandrakeRoot]

Mathematics is not an empirical science. It is based on logical deduction. When doing applied mathematics it is important to verify your results with observations.

 

It is true that physical laws have restrictions on their domain of use. In pure mathematics restrictions are clearly formulated in the theorems and lemmes. When applying math to the "real world" (whatever you may mean with that) it is up to you to verify that these conditions are satisfied. (Like in your example above that indeed all scientists are aliens). The fact that not all scientists are aliens does not in any case make untrue the conclusion "Einstein is a scientist => Einstein is an alien", under the hypothesis that all scientists are aliens.

 

Mandrake

[Daymare17]

i disagree.. imo every person has his own reality. his own perception of things. no brain works exactly the same as any other. if you take mind altering substances like alcohol e.g. then your reality is different. if you take LSD then your reality and perception etc also changes. if you stop breathing you get dizzy .. see what i mean?

 

But even if you take alcohol or LSD, you will fall down the stairs if you trip. Even though a primitive tribesman may not know of Newton's theory of gravitation, that doesn't mean he can jump off a cliff and get unscathed away with it. See what I mean? The world is objective, it exists independent of our perception of it.

[Daymare17]

Mathematics is not an empirical science. It is based on logical deduction. When doing applied mathematics it is important to verify your results with observations.

 

It is true that physical laws have restrictions on their domain of use. In pure mathematics restrictions are clearly formulated in the theorems and lemmes. When applying math to the "real world" (whatever you may mean with that) it is up to you to verify that these conditions are satisfied. (Like in your example above that indeed all scientists are aliens). The fact that not all scientists are aliens does not in any case make untrue the conclusion "Einstein is a scientist => Einstein is an alien"' date=' under the hypothesis that all scientists are aliens.

 

Mandrake[/quote']

 

But you can't verify applied mathematics, since x not= x.

[Sayonara]

Judging from what I see, if mathematics is what you pass off as "mathematics", then Platonism is not just popular in mathematics - it IS mathematics! Plato was a pure idealist. If you had a circular plate, for instance, according to him it was not a plate, it was just an unperfect expression of the Ideal Circle. This stands reality completely on its head.

Isn't that a peculiarity of Plato, rather than a peculiarity of mathematics?

[fairychild]

But even if you take alcohol or LSD, you will fall down the stairs if you trip. Even though a primitive tribesman may not know of Newton's theory of gravitation, that doesn't mean he can jump off a cliff and get unscathed away with it. See what I mean? The world is objective, it exists independent of our perception of it.

nope.. definately not mate. there is no absolute reality from which u can derive an objective viewpoint. einstein in his first paper banished the concepts of absolute time and absolute space. there aint such a thing. if you move, your time will go different than someone else who stands still.. etc

[matt grime]

But you can't verify applied mathematics, since x not= x.

 

 

If you can give us one single mathematical object which is not equal to itself, in the correct context of equal, then perhaps we all should stop. However all you've offered is that an apple, or some other physical object, is not equal to itself, with some odd sense of equal.

 

Consider the number sqrt(2), consider the smallest positive root in the reals to the polynomial x squared minus two. In what *mathematical* way are they not equal?

 

Incidentally, the mathematical philsophy that prevails currently appears to be formalism, not idealism, or platonism.

 

What ever the initial motivation to study was doesn't mean we shouildn't go beyond it. And you're still wrong about the motivation for 1=1.We are not saying physical objects are equal but the (well defined quantity when it exists) of two collections are equal. Ie if we started counting things one at a time in each collection starting at 1 we'd hit the same number when we came to the final object in each collection. That is all. We are not claiming anything like the objects are equal, so stop saying we are.

[Daymare17]

nope.. definately not mate. there is no absolute reality from which u can derive an objective viewpoint. einstein in his first paper banished the concepts of absolute time and absolute space. there aint such a thing. if you move, your time will go different than someone else who stands still.. etc

 

It was a juvenile mistake of his, which he honestly admitted in his later years. He was influenced by the empirio-critical philosophy of Ernst Mach. It's very sad that so much of 20th century physics is based on this error of Einstein, which he himself repudiated.

[fairychild]

It was a juvenile mistake of his, which he honestly admitted in his later years. He was influenced by the empirio-critical philosophy of Ernst Mach. It's very sad that so much of 20th century physics is based on this error of Einstein, which he himself [b']repudiated.[/b]

..?? so u wanna say time and mass is always constant?

[MandrakeRoot]

But you can't verify applied mathematics, since x not= x.

 

You still dont see the difference between application and theory. It is hopeless.

 

Mandrake

[haggy]

And who's Doron Shadmi anyway?

Someone whose reponses are quite similar to yours.

 

A typical conversation with Doron (*must spell correctly so it's not deleted*) goes something like this:

Doron: "I see things a particular way and in that context the whole of mathematics must be incorrect"

 

Respondent (typically Matt Grime): *A sound argument refuting what Doron had to say*

 

Doron: "You just don't get it(condescending tone)' date=' you've got to look at the way I do!"

.

.

.

More sound arguments

.

.

.

Doron: "By my dodgy vague arguments all of Mathematics is wrong so you had better change your view people!"

[/quote']

[MandrakeRoot]

Yeah that is pretty much a typical discussion with Doron :=)

 

Mandrake

[matt grime]

Doron's arguments maybe sound in and of themself, though I seriously doubt it as no one has ever managed to make him explain anything of it in terms that are sound. I object to his presumptions of redefining (though without explicitly saying as what) terms that are well known, and doing something unmathematical with it based upon some of his personal preferences, to conclude that maths itself is flawed. Typical example: Cantor's diagonal argument is wrong since there is no such thing as an infinite "complete" set, where "complete" is not defined. The clincher in this is that in the alleged refutation of it he actually gives the original proof and doesn't manage to demonstrate where it is wrong. Everyone should remember, especially Daymare, that there is no such thing as absolute truth in mathematics.

 

And if you don't believe that, D17, then can I suggest you examine Skolem's paradox - that there is a model of ZF in which all sets (including the power set of N) are countable. Countable though in the sense that although there is a bijection with N, that bijection may not be set in the model (and in the case of the power set, which isn't the power set you're thinking of, certainly isn't - Cantor's argument is still true). Of course I doubt you'll believe that functions are sets.

[premjan]

well universal quantifiers show up even in the definition of a limit (for all epsilon, there is a delta or N for which...) so in a sense, under constructive math, it is not sound to speak of such a definition for a limit.

 

Strictly speaking, infinite decimals ought not to be allowed at all since they are not constructible, so Cantor's proof would fall apart if we were forced to use only constructible numbers.

 

Thing, is, strict constructability is more of computer science than math, and that is why even Doron doesn't push the argument home, since he is clearly coming more from a logic than a CS angle.

[matt grime]

"Strictly speaking, infinite decimals ought not to be allowed at all since they are not constructible, so Cantor's proof would fall apart if we were forced to use only constructible numbers."

 

In what sense are you being strict? Since any constructible number has an (infinitely long) decimal expansion, this is a very moot point. However, you are almost making the same mistake as Doron. Cantor's argument isn't about constructible numbers so your objections are off topic, and slightly irrelevant. We aren't forced to use constructible numbers, nor shoud we be.

[Daymare17]

And if you don't believe that, D17, then can I suggest you examine Skolem's paradox - that there is a model of ZF in which all sets (including the power set of N) are countable. Countable though in the sense that although there is a bijection with N, that bijection may not be set in the model (and in the case of the power set, which isn't the power set you're thinking of, certainly isn't - Cantor's argument is still true). Of course I doubt you'll believe that functions are sets.

 

I'm not a mathematician so can you explain in less professional terms exactly what you mean?

 

If you can give us one single mathematical object which is not equal to itself, in the correct context of equal, then perhaps we all should stop. However all you've offered is that an apple, or some other physical object, is snot equal to itself, with some odd sense of equal.

 

Consider the number sqrt(2), consider the smallest positive root in the reals to the polynomial x squared minus two. In what *mathematical* way are they not equal?

 

I don't have to consider any mathematical object, since all mathematical objects are abstractions of real objects and need to be accountable to them in some way. You can "prove" anything on its own terms. You're simply evading the question. Mathematics has serious problems with describing reality and you seem to think that this is well and good and should remain that way. You think that maths does not have to describe reality but stands fine on its own legs. In this you are joined by many of today's theoretical physicists, by the way. They start with some vast equations, and then select those facts that they think look good with the equations, or in the worst cases no facts at all. Then they pass this jumble off as a hypothesis. It's very poor science. Maths actually arose precisely from describing reality. Are you religious? No? Well, then how can you say that mathematics has existed since the beginning of the universe? Since that is what you imply when you say that it is independent of the real, material, human world.

 

Your view is an unfortunate consequence of the extreme division of labor.

 

Incidentally, the mathematical philsophy that prevails currently appears to be formalism, not idealism, or platonism.

 

But formalism is idealism, in the broad sense. Idealism means the idea that matter comes from ideas. That's what you say when you say that the apple is equal to itself. Formal logic states that x = x. But x is always changing. So it's never x. Thus, you're arbitrarily foisting an idea (which is more or less true, but not completely true) upon the real world. You're committing violence to reality.

 

What ever the initial motivation to study was doesn't mean we shouildn't go beyond it.

 

I never said that. I explicitly said that I have no gripe with abstract maths as such. The greater the abstractness of the mathematics, the greater its ability to describe real interrelationships between things - but also the greater is the danger of one forgetting its material roots.

 

And you're still wrong about the motivation for 1=1.We are not saying physical objects are equal but the (well defined quantity when it exists) of two collections are equal. Ie if we started counting things one at a time in each collection starting at 1 we'd hit the same number when we came to the final object in each collection. That is all. We are not claiming anything like the objects are equal, so stop saying we are.

 

"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."

 

I said that x not= x. That means that the apple is not equal to itself, since it's always changing. It's never equal to itself. Never! It's a wrong conception of your mind.

[premjan]

"Strictly speaking' date=' infinite decimals ought not to be allowed at all since they are not constructible, so Cantor's proof would fall apart if we were forced to use only constructible numbers."

 

In what sense are you being strict? Since any constructible number has an (infinitely long) decimal expansion, this is a very moot point. However, you are almost making the same mistake as Doron. Cantor's argument isn't about constructible numbers so your objections are off topic, and slightly irrelevant. We aren't forced to use constructible numbers, nor shoud we be.[/quote']

I guess I meant numbers that have finite representations ought to be usable, those that don't should not...but I guess most (transcendental) numbers are not finitely representable, is that right? And depending on your starting point (e.g. if we measure numbers in multiples of pi, then some transcendental numbers become representable.).