# x not= x

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To be honest I neither know nor care about constructible numbers. I was merely pointing out that in saying Cantor's argument doesn't apply to them you are making a misleading statement since all cantor does is show that there is no bijection between a set and its power set.

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but the diagonal argument does this only for Z and R.

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I'm not a mathematician

Gee, I think we all find that surprising

You ramble on about how Mathematics has to conform to your changing apples and how x != x.

If you denote "The exact state that an apple is in" by x (if that were quantifiable in some manner), this does not imply that x != x. Your changing apple (no matter what the time) does not mean this in the slightest. What it would rather mean is that your apple is in a new state x' and not its original state. The best you could say is that x != x'.

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but the diagonal argument does this only for Z and R.

No it doesn't:

Given any set S, let f be an injection from S to P(S), its power set. Define T to be { t in S : t not in f(t)} then T is not in the image of f - and this is more in the spirit of how Cantor wrote his proof originally (and is only one of few of the proofs he offered).

That is all the diagonal argument is - there is nothing special about N and R in it, other than there is a nice explicit description that shows us how it works.

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Doron claims that Dedekind cuts do not work because the limit point of a sequence (e.g. sequence of rationals) is always separate by a finite interval from its "limit" irrational (and due to the self-similar scaling of the real line, any gap is a "1" gap.

Here's his latest set of posts:

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Yes, because I predict he's completely ignoring the definitions and using his own private meanings

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Doron claims that Dedekind cuts do not work because the limit point of a sequence (e.g. sequence of rationals) is always separate by a finite interval from its "limit" irrational (and due to the self-similar scaling of the real line' date=' any gap is a "1" gap.

Here's his latest set of posts:

Doron always talks a lot, i wouldnt pay attention to his pseudo-intuitive arguments.

Mandrake

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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.

You're very confused.

1) Formalism is not idealism in any sense (not even in a very very very broad sense, whatever "broad sense" means). To say that it is suggests that you have no idea what you're talking about. Until you can prove otherwise I'll assume that to be true; i.e. that you have no idea what you're talking about.

2) I don't get why the "=" confuses you. 1+1=2 is very simple as long as you understand the "+" and the "=". Maybe you can't count.

3) As hard as it is for you to understand that 1 is 1 ... 1 is 1. No matter which side of the "=" it's on 1 is the same number.

4) The same goes for x. x is x. No matter which side of the "=" x is on it's the same letter.

5) Maybe the problem here is that you're not trying to understand a General Mathmatics problem, but trying to come to terms with reality; i.e. define reality for yourself philisophically. If so, this is a General Mathematics thread. The wrong place for you.

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Yes or no: The origins of mathematics is the real world.

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Yes or no: The origins of mathematics is the real world.

One of the origins of scientific method was with trying to find a philosophical validation for god. That doesn't mean that science is religion.

Mathematics, and logic, are entirely abstract entities. ENTIRELY. Whatever issues you may have with their application, they are not based in measurement, but on axiom. Mathematics is correct for its set of axioms. If you disagree with one of the axioms, then you're just making a different mathematics.

Oh, and re- apple =! apple... what if the only property you are measuring is that it can be referred to as an apple? Surely then any given apple = any other, or indeed the same, given apple, since the only properties you can compare are that they are, in fact, apples?

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.

Space and time are not absolutes. This has been empirically proved over and over and over again.

Daymare, you're arguing with invalid technical and philosophical arguments.

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• 2 weeks later...

I have here a piece that I hope many of the mathematicians will read, if nothing else than to get their discipline described from different viewpoints.

Does Mathematics Reflect Reality?

Extract:

"The content of "pure" mathematics is ultimately derived from the material world. The idea that the truths of mathematics are a special kind of knowledge that is inborn or of divine inspiration does not bear serious examination. Mathematics deals with the quantitative relations of the real world. Its so-called axioms only appear to be self-evident to us because they are the product of a long period of observation and experience of reality. Unfortunately, this fact seems to be lost on many present-day theoretical mathematicians who delude themselves into thinking that their "pure" subject has nothing to do with the crude world of material things. This is a clear example of the negative consequences of carrying the division of labour to the extreme.

From Pythagoras onwards, the most extravagant claims have been made on behalf of mathematics, which has been portrayed as the queen of the sciences, the magic key opening all doors of the universe. Breaking free from all contact with the physical world, mathematics appeared to soar into the heavens, where it acquired a god-like existence, obeying no rule but its own. Thus, the great mathematician Henri Poincaré, in the early years of this century, could claim that the laws of science did not relate to the real world at all, but represented arbitrary conventions destined to promote a more convenient and "useful" description of the corresponding phenomena. Certain theoretical physicists now openly state that the validity of their mathematical models does not depend upon empirical verification, but on the aesthetic qualities of their equations.

The theories of mathematics have been, on the one side, the source of tremendous scientific advance, and, on the other, the origin of numerous errors and misconceptions which have had, and are still having profoundly negative consequences. The central error is to attempt to reduce the complex, dynamic and contradictory workings of nature to static, orderly quantitative formulae. Nature is presented in a formalistic manner, as a single-dimensional point, which becomes a line, which becomes a plane, a cube, a sphere, and so on. However, the idea that pure mathematics is absolute thought, unsullied by contact with material things is far from the truth. We use the decimal system, not because of logical deduction or "free will," but because we have ten fingers. The word "digital" comes from the Latin word for fingers. And to this day, a schoolboy will secretly count his material fingers beneath a material desk, before arriving at the answer to an abstract mathematical problem. In so doing, the child is unconsciously retracing the way in which early humans learned to count."

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The moron is back

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The excerpt is riddled with errors as anyone who understands, say, the continuum hypothesis will know. It seems based upon a high school idea of mathetamatics. In particlar Tarski has shown that geometry is complete, thus it cannot be quantitative, if by that we mean strong enough to necessasrily define the integers.

That is of cousre on problem with these debates - what exactly do you mean by quantitative?

Moreover, how many real pure mathematcians do you know? I'm sure a lot of them would like to know what on earth the decimal system has to do with this argument, cos I'm buggered if I can see the author's point.

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Have read the article now, or at least a large chunk of it before I gave up in digust at the ignorance of the author. Do you want a list of the mistakes in it, Daymare?

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Moreover, how many real pure mathematcians do you know? I'm sure a lot of them would like to know what on earth the decimal system has to do with this argument, cos I'm buggered if I can see the author's point.

I don't know any real mathematicians, I'm a total amateur. But I think that even an amateur can see many of the obvious contradictions that abound in mathematics.

The article is an answer to the general idea that mathematics has nothing to do with reality. Here is a pure specimen of this mystifying thinking:

"One of the origins of scientific method was with trying to find a philosophical validation for god. That doesn't mean that science is religion.

Mathematics, and logic, are entirely abstract entities. ENTIRELY. Whatever issues you may have with their application, they are not based in measurement, but on axiom. Mathematics is correct for its set of axioms. If you disagree with one of the axioms, then you're just making a different mathematics (JaKiri)."

JaKiri, like most of the mathematicians here, is essentially a Pythagorean number-worshipper. So as you see, the article is quite relevant.

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"One of the origins of scientific method was with trying to find a philosophical validation for god. That doesn't mean that science is religion.

Mathematics, and logic, are entirely abstract entities. ENTIRELY. Whatever issues you may have with their application, they are not based in measurement, but on axiom."

The article makes the point that axioms in turn are based on measurement, on reality. They don't drop from the sky like certain persons seem to think.

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I don't know any real mathematicians, I'm a total amateur. But I think that even an amateur can see many of the obvious contradictions that abound in mathematics.

Since you're an amateur, and the mathematicians are telling you that you are wrong, don't you think "I simply don't understand why they are not contradictions" might be the more likely solution?

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So what about the fact that, for instance, the whether or not we take the parallel postulate as true or false in geometry (it is independent of the other axioms) leads to 3 totally different theories all of which are useful in the "real world"? Isn or isn't this one of the inherent problems with mathematics. When we say it is `just' a study of results that may be derived from axioms that we may assume true or false (as long as we're not inconsistent) we are not claiming anything about the absolute truth of them at all.

Do you think that the axiom of choice is in some physical sense? Or false? Please give me a physical model to verify this.

You didn't answer my question about what the arbitrary choice of the number 10 as our radix has to do with the philosophical issues at hand. Nor about Tarski's completeness theorem. You do understand what that is? After all you are presuming to tell us what we ought to think about what mathematics is, so I'd presume you're familiar with this statement. I must admit I know very little about O-minimal structures (very very little indeed.)

Moreover, I trust you don't believe the article's assertion that base 10 and base 20 are the only systems developed by 'ancient people' or whatever sill phrase they used.

I could also take issue with the entirely unjustifed premise that mathematicians and scientists are attempting to abuse geometry (reduce nature to points etc) to create an ontological explanation for the nature of the universe. We do not. We create models with the best consistent tools we have and use them to see if they are good enough MODELS.

No one will deny mathematics' origin, or that for many years it was developed intuitionally. However the author has this point completely backwards, what the author fails to realize is that it is precisely the division of mathematics from its origins that has led to great advances and cross fretilization of ideas in many disciplines.

Particularly sickening is the implied statement that imgainary numbers are some how bad by placing the name in quotation marks. Agreed the name is misleading, not to say downright unhelpful, but they are nto a matter for distaste like that. As a standard challenge I like to ask people who have philosophical issues with them to define the real numbers. Usually they cannot, since if they can they know that complex numbers are just ordered pairs of reals with an algebraic structure that makes them a field.

We could also make the point that, along with many people, the author (and by extension you) might not fully understand what a mathematician means when they say theirs is an abstract subject: they mean that it is not *necessary* to refer to any underlying physical situation that may have led to something being studied, even if there were one. Look at the transition from abstract to concrete groups.

For instance, what is the quantitative meaning of the axiom: given any line and a point not on that line there is a line passing through that point and not intersecting the initial line (where by line we mean geodesic)? What is quantitative about saying that x_n is a homoclinic orbit (apart from the fact we indexed it by something you think is a number, though I could easily, if inelegantly, have witten it in set terms). Or how about if G is a group and H a subgroup then any cosets gH and kH intersect trivially or are the same.

To make a wholly crass analogy, language (we may posit) was devised to express basic desires and needs, such as to say we are hungry, and explain how to get food. Should we have to remember this when reading, to quote on critic, the saddest tale ever told: "The good soldier" by Ford Maddox Ford/Hueffer.

But thank you, a self admitted neophyte, for pointing out an article containing errors factual and possibly philosophical. I shall try and remember that when I'm sat in the lectures in my next conference: epistemiology and philosophy of mathematics, next month.

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hey.... are we talking about the apples as individuals ,or apple as analogy of quantity?

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i think where someone went wrong was when they tried to represent something theoretical with something physical. Of couse no two apples are perfectly similar, but it is the value that each represents that plays into the equation.

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Dear Daymare17,

There is more than one school of thought in the world of Mathematics.

The main schools of the modern time are Formalism and Logicism.

But as you say, things can be changed, and we hope that the changes are for the better.

If you ask me, then after all, all of us including our abstract thoughts belong to the changing reality.

One of the important things in the language of Mathematics is the ability to share ideas in a very accurate way between different persons around our planet.

In order to do this, we have to ignore the changing reality and stick to unchanged models, which can be easily understood and developed by the people which share them.

So, first you find that the time concept does not exist in this way of thinking, which is called logical reasoning.

There are advantages in this timeless way of thinking, because they can help us to take some perspective point of view from changing reality, and maybe discover some deep connections that maybe or maybe not can be translated to the changing reality, but these deep connections can help us to organize our abstract/non-abstract world in cleverer, efficient, beautiful and unexpected ways, which can change our (at least) near reality.

I agree with you, that if we go too far by totally ignore the non-trivial complexity of the changing reality, by using mostly a False_XOR_True (0 either 1) way of thinking, then there is a real danger that we become a closed community of scholars that all they do is to develop methods to share their scholastic ideas, where the ideas themselves are not developed beyond the 0 xor 1 way of thinking, which is defiantly a trivial model, when it is compared to the non-trivial changing reality.

The keyword, in my opinion, is to find balanced gateways between the abstract worlds of our trivial models, and the non-trivial changing reality.

It means that from time to time our most fundamental abstract concepts like (for example) the Number concept must be re-searched, re-examined, re-discussed, re-thought and understood from new points of view, which are based on new insights that are created/discovered in both abstract and non-abstract word.

For example, concepts like redundancy and uncertainty that are coming form quantum phenomena, information theory, chaos, etc... can be used as first-order properties of the Number concept, and as the result, we can get a much more comprehensive framework, that can help us to develop abstract/non-abstract worlds, which are less trivial than the standard Black_XOR_White approach.

In order to see some example of this approach please look at:

http://www.geocities.com/complementarytheory/TheBestOf.pdf

Yours,

lama

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JaKiri, like most of the mathematicians here, is essentially a Pythagorean number-worshipper. So as you see, the article is quite relevant.

The article makes the point that axioms in turn are based on measurement, on reality. They don't drop from the sky like certain persons seem to think.

You're misinterpreting the post. What you're missing is that if we took our mathematical axioms to another universe with completely different physical rules, then the mathematics would be identical.

Frankly, that's a decent counterargument to article as well.

List of problems with article:

1. Certain theoretical physicists now openly state that the validity of their mathematical models does not depend upon empirical verification, but on the aesthetic qualities of their equations.

Certain theoretical physicists? Who?

2. Confusion of mathematical modelling with mathematics.

If a scientific equation is incorrect, it's not because of 'mathematics' causing simplification. It's due to that equation being correct to the limitations of our measuring capability. Of course, that's still irrelevent, as that's physics (et al), not mathematics.

3. Confusion of material derivation of certain axioms with independence from material derivation.

For example, the base 10 counting system mentioned in the article. It makes not a jot to mathematics how we count; we could use base 93 and the mathematics would all be identical.

4. The misquoting of Aristotle to try to get across a point.

The article says Aristotle:

"The mathematician investigates abstractions. He eliminates all sensible qualities like weight, density, temperature, etc., leaving only the quantitative and continuous (in one, two or three dimensions) and its essential attributes."

Aristotle said:

As the mathematician investigates abstractions (for before beginning his investigation he strips off all the sensible qualities, e.g. weight and lightness, hardness and its contrary, and also heat and cold and the other sensible contrarieties (and so on for quite some time).

The article's version implies that Aristotle thought that the mathematician removed everything sensible, whereas the actual quote implies that the mathematician removes everything that it is sensible to have removed.

I can't actually find any basis for the second quote. If you find it, please tell me, because it goes against what philosophy of Aristotle I have read.

5. Strawman: The first

It's not part of mathematical theory that counting systems (or other entities) were not derived from reality, merely that they are independent of it. Six paragraphs are devoted to this Strawman, so it should probably be counted as more than a mere single example. But, on we go.

6. Engels was not a mathematician, and had little more than a basic grounding in mathematics.

In an appeal to authority, it seems irrational to choose an authority who has little grounding in the subject. If only the author of the piece had chosen one of the mathematicians who shared Engels's opinions! If only one existed!

7. Thus we have irrational numbers, imaginary numbers, transcendental numbers, transfinite numbers, all displaying strange and contradictory features

Whilst some people may find some of the above strang, they are not contradictory. Another paragraph wasted! In fact, this is a whole wasted section, because it is spent entire on this flawed argument.

8. Making a logical problem out of Zeno's paradox.

From the article: '"This paradox still perplexes even those who know that it is possible to find the sum of an infinite series of numbers forming a geometrical progression whose common ratio is less than 1, and whose terms consequently become smaller and smaller and thus ?converge? on some limiting value."'

This just isn't true. I know many people who aren't perplexed by this explanation of it, and using qualitiative evidence in a mathematical discussion is fairly bad form. There are a multitude of other explanations for the 'paradox', as well.

9. Physics is not mathematics

'Modern physics accepts that the number of instants between two seconds is infinite, just as the number of instants in a span of time with neither beginning nor end.'

This is incorrect. Modern mathematics says that, between any two given points, there are an infinite number of points.

Modern physics has quanta of length and time.

10. Everything in the section on infinity is rubbish

It really is. I was going to make another few points on this section alone, but it really isn't worth the bother, because they all make the same basic mistake: the article thinks that infinity causes a problem to modern mathematics.

It doesn't.

11. Oh wait, one more from 'infinity'

'This immediately leads us into a logical contradiction. It contradicts the axiom that the whole is greater than any of its parts, inasmuch as not all the positive integers are perfect squares, and all the perfect squares form part of all the positive integers.'

Hey guys! Inventing axioms is fun!

12. The Calculus

This isn't a problem per se, I'm just confused as to what the point of the entire section is.

13. Crisis In Mathematics

I'm getting bored of the constant misrepresentation in this piece. I'm going to spend the rest of these comments quoting from popular Cartoons of the last 20 years.

14. Mathematics isn't physics: part II

Bond... James Bond... I'll do it!

15. THAT ISN'T WHAT CHAOS IS YOU FOOLS! AAARGH YOU'VE MANAGED TO SOMEHOW COMPLETELY MISINTERPRET THE RESULTS OF LORENZ'S FINDINGS AND NOW I'M GOING TO HAVE TO FIT THE EXPLANATION OF WHY THIS IS INCORRECT INTO THE HEADER BECAUSE, BY MY OWN RULES, I CAN'T PUT IT IN THE FOLLOWING COMMENTARY! LORENZ DIDN'T FIND THAT, FROM THE SAME BEGINNINGS, YOU'D GET DIFFERENT RESULTS! HE FOUND THAT, FROM EVER SO SLIGHTLY DIFFERENT BEGINNINGS, YOU GOT RESULTS WHICH STARTED SIMILARLY THEN DECREASED IN SUCH OVER TIME DUE TO THE POSITIVE FEEDBACK EFFECT OF THE MATHEMATICAL SYSTEM HE WAS MODELLING! SWEET JESUS, HOW DID THE WRITER OF THIS ARTICLE COME UP WITH SUCH TRIPE?

I am ZIM!

16. I actually stopped reading at this point. I'm on the verge of tears because of your stupid article. Not really.

Do not cross me, I control your arms!

Here's something which may come in useful...

http://www.datanation.com/fallacies/index.htm

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As for the 1 does not equal 1 disproof, here is a very simple equation:

z=x+y, thus x=z-y, substitute z=z-y+y, z=z. However this is true under the condition that z=z, x=x, and y=y. If that condition is not true, thus the axiom of x=x not true, then z may or may not equal z.

The reason there is no way to prove or disprove this is that this is the fundamental axiom of mathematics, and the only way to disprove that z=z as our friend Daymare here is proposing is to assume that z=z thus creating a contradiction as to his proof.

There is no way to prove the axiom either, and that's why pure mathematics is USEFUL if it is applied. That's the proof, physical proof, but it's not mathematical proof, and thus it is an axiom. I think there are systems where x does not equal x, and such axioms are unprovable too. Such as 10/10=1/2, this statement cannot be proven true or untrue.

Whether a nature can be based on such a rule is under the extent of your imagination, since it could have some arbitrary value, say "v" meaning that all of it exists if and only if only half of it exists as the statement 10/10=1/2 suggests

In pure mathematics these two systems are both valid under their axioms, and there's no need to ridicule a person who's seeing the fact that axioms are unproven mathematically. It is only in nature that they apply.

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You should at least point out that you are not using the convention definition of 10, 2, 1 or / in that example.

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