Please show us how the limit concept is rigorous

[matt grime]

Do you know what proof by contradiction means? We entertain a silly idea temporarily putting aside our intuition and show that there is a genuine contradiction.

What we do here is to prove A=>B is to show that not(B) => not(A)

we've now omitted the step you find objectionable entirely by proving that it is not possible.

So let us take what I had in my previous post: would have to have. Do you understand what that means?

We are showing that |a-b| not zero implies that (since |a-b| <|a-b| is impossible) that they hypothesis is impossible. That is what we want, because it shows the negation of the conclusion implies the negation of the hypotheses, thus the hypotheses imply the conclusion.

This is very simple logic and every time you post indicating you don't understand the proof you are weakening yet more your stance as being some ultimate arbiter of mathematical correctness.

I wonder, how long do you actually take to try and understand the answers you get?

 

[Organic]

|a-b| <|a-b|

1) a not= b

2) abs(a-b)=d < e > 0 < d

Please show how by (1) and (2) you can prove by contradiction that d = 0

 

[pig]

if d is smaller than any possible number > 0, then d cannot be > 0 because it would have to be smaller than itself. what is so hard for you to understand here?

 

[Organic]

if d is smaller than any number > 0, then d cannot be > 0 because it would have to be smaller than itself

No, abs(a-b)=d < e > 0 is smaller then any number accsept 0, because d approaching but never reaching 0.

I you can't understand that simple thing?

 

[matt grime]

Occam's razor: two seemingly eloquent expositors of mathematics say you are wrong, you insist you are right without being able to explain yourself coherently... hmm, wonder how that's going to work out.

We've omitted any mention of e, and e<d and all the things you found unacceptable so what else are we supposed to do?

We are saying that it is impossible for any real strictly positive number to be less than every other real strictly positive number, that's all.

It's not very hard, and it doesn't even mention e or d.

 

[pig]

No, abs(a-b)=d < e > 0 is smaller then any number accsept 0, because d approaching but never reaching 0.

I you can't understand that simple thing?

wait a minute. you say it is smaller than any number except 0, but is "never reaching 0". what is it then? santa claus? i am wasting my time here.

 

[matt grime]

but d isn't approaching anything because d is fixed, organic.

 

[Organic]

Matt,

You are looking on your impossible d from the wrong side.

Look at d from 0 side and see how d always > 0, because there are infinitely many of them between any given d and 0.

Shortly speaking, if d=0 then we have the smallest d, and as we know, this collection of infenitely many elements > 0 has no smallest element.

 

[pig]

please demonstrate this on the example of a=2, b=2. what is there between d and 0?

 

[matt grime]

that could be a goody...

d is |a-b| and a and b are fixed not mention the fact thta card((0,d)) isn't important in the slightest, really (not that we expect you to realize that and not that you even know what card means either).

 

[Organic]

please demonstrate this on the example of a=2, b=2

a not= b, this is the first rule.

 

[pig]

a not= b, this is the first rule.

sorry, i didn't realize we were still stuck there, remind me what exactly are you trying to prove? just please be clear and exact.

i mean, i now have a feeling you're trying to prove that if d>0, then d cannot be 0, which is self-evident.

 

[Hurkyl]

Since we've left the domain of mathematics, we're also leaving the domain of the mathematics forum.

 

[matt grime]

were we (well organic) ever in the realm of mathematics?

 

[Organic]

Matt,

not that we expect you to realize that and not that you even know what card means either

This is the whole idea.

a not= b --> 1=|{d}|

a = b --> 0=|{}|

So as you see to reach 0, d has a phase transition form 1(exists) to 0(does not exist).

By this phase transition we are breaking the rules by eliminating d existence.

And why we are breaking the rules? because our case is the Math universe where a not= b.

When a=b we made a phase transition to another Math universe where d does not exist, therefore cannot be used in our proof as the impossible d<d.

if d is smaller than any possible number > 0, then d cannot be > 0 because it would have to be smaller than itself. what is so hard for you to understand here?

1) a not= b

2) abs(a-b)=d < e > 0

d is always smaller than e where e>0, and we never comparing d to itself but only to e.

Therefore we can never conclude that d<d.

 

[matt grime]

crank the crackpot index by a couple more points.

 

[pig]

Therefore we can never conclude that d<d.

we conclude that a=b, because otherwise d would be < d which is impossible. what do you find wrong with my statement that you quoted? i am trying to avoid mentioning "e" because you clearly do not understand what it represents from the very beginning.

 

[Organic]

if d is smaller than any possible number > 0, then d cannot be > 0 because it would have to be smaller than itself. what is so hard for you to understand here?

1) a not= b

2) abs(a-b)=d < e > 0

d is always smaller than e where e>0, and we never comparing d to itself but only to e.

Therefore we can never conclude that the impossibility of d<d --> a=b.

 

[matt grime]

But that isn't what we conlcude at all

 

[Organic]

Therefore we can never conclude that the impossibility of d<d --> a=b.

 

[matt grime]

If we show you a stick, are you guaranteed to get the wrong end of it?

 

[pig]

organic, here is a simpler but related problem written in a way that should be familiar to you.

Little Johnny is the best mathematician in his class. One morning, during the break, the teacher decides to test his logical skills. She knows Johnny often mentions how he likes all numbers which are larger than 0, and only those. When she waves at Johnny, a candy bar in her hand, he runs up to her desk within seconds. "Johnny, dear, I am thinking about a number. You wouldn't like this number, it is too small, but it is not negative. If you guess the number, the candy bar is yours. If you guess wrong, the fat kid in the third row gets it." Johnny's answer was zero.

Who got the candy bar?

A) Johnny
B) The fat kid in the third row
C) Eric Cartman
D) All of them approach, but never reach the candy bar
E) The candy bar was phase-transfered to another candy bar universe because it broke the rules

 

[Organic]

Pig, it is a beautiful story!

I am not so good in English, but I'll try my best.

One morning the farmer said to his last pig in the farm:

Here is a sand clock.

After the last grain of sand felled down you will be eaten.

Pig loved life very much, so he prayed to the god of mercy

The god of mercy heard his pray and came to help him.

He looked on the sand clock and said:

"Do not be afraid little pig, I have here another sand clock, that from outside looks exactly like the farmer's sand clock but inside of it time never ends, because this is the send clock of eternity of gods.

But little pig, you have to find by yourself how it works, otherwise I cannot help you, all I can tell you is that one and only one grain is falling down to the bottom of the sand clock each time ."

The pig knew Math and he loved very much irrational numbers, so he thought to himself:

What if each grain of sand is like some irrational number that its right side is always open, and it means that it has no exact value.

It means that I can use this open side to find how the sand clock of eternity works.

So he said to the god of mercy:

"My answer is this:

Each time when a one grain of sand is falling down to the bottom of the sand clock, each grain in the top side of the clock is divided to two pieces, and the width of the entrance between the top and the bottom sides of the send clock, becomes 1/2 of its previous width.

In this case our sand clock is the clock of eternity."

And the little pig got his life back and enjoyed every piece of grain of it, until his last day.

And before he died he discoverd the secret of the clock of eternity to his sons, and they discovered it to their sons, and they discovered it to their sons, and they discovered it to their sons, and they discovered it to their sons, and they discovered it to their sons, and they discovered it to their sons, and they discovered it to their sons, and they discovered it to their sons, and they discovered it to their sons, and they discovered it to their sons, and they discovered it to their sons ...


(By the way, from my point of view 0 is not positive AND also not negative).

 

[matt grime]

presumes real space and so on is infinitely divisible, which it isn't. See what happens when you think that physical inutuition and mathematical intuition are the same thing?

 

[Organic]

Yes Matt, my intuition is the theoretical one, your is the physical one where infinitely many elements has fixed properties, and we can use words like "all" or "complete" that related to them.

On the contrary, the theoretical infinity (what I call potential infinity) is an open collection, which incompleteness is one of its fundamental properties.

And the reason that in your world 0 is a posivite number, based on the excluded middle idea of the Boolean or Fuzzy Logics.

My intuitions are based on Complementary Logic:

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

Boolean or Fuzzy Logics are proper sub-systems of Complementary Logic, therefore They are going to get off stage as main logical systems.

 

[matt grime]

you aren't making sense. In fact you've entirely switched your view that I am a thoerist just interested in definitions and it was you that was dealing with the acutal reality of mathematics as a physical entity.

Your second paragraph starts as though you are going to explain why I think something, and then doesn't explain anything. (and whether or not you declare 0 to be positive or negative or neither or both is personal preference)

 

[Organic]

and it was you that was dealing with the acutal reality of mathematics as a physical entity.

What I showed is that actual infinity is too strong or too weak to be modulated by using Math language.

More details can be found here:
http://www.geocities.com/complementarytheory/Theory.pdf

Your second paragraph starts as though you are going to explain why I think something, and then doesn't explain anything.

Between us, please show me a one case where you tried to understand me.

 

[matt grime]

Well you could take that second paragraph and rewrite it so that it makes sense, in some sense. For instance if you replaced the comma with 'is' it might make some sense, but would still be inaccurate.

Why don't you demonstrate that at some point you've attempted to understand any of the very basic mathematics upon which you've seen fit to pronounce judgement despite obviously being incredibly ignorant and idiotically stupid?

 

[Organic]

Even if math originated solely from our brain it would still remain an underlying principle of nature.

Do you think that our brain totally-included, half-included or totally not-included to the underlying principle of nature?

Can you demonstrate an abstract thought which is totally not influenced by reality?

Do you think that professional mathematicians can develop Math in one hand but on the other hand they say: "We don't care about reality when we develop our definitions"?

For example:

Speaking up for (some of ) the mathematicians: we don't care. If we did we'd be doing philosophy. Would the martians have derived that equation? Perhaps, perhaps not - they almost certianly wouldn't have devised the same way of presenting it, and we couldn't tell if they'd picked i or -i as their square root of -1, which they may have called something else anyway. That answer has a superficial and a non-superficial part to it.

There is a "philosophy" section to Physics forum and this probably belongs there.

Can we ignore our abilities to develop Math language by saying that our abilities to develop Math is not mathematical but a philosophical question?

Please be aware that not some of but most of the professional mathematicians have Matt's opinion on these questions, and the reason is very simple, beside learning Math in the universities they also learn from their teachers that the logical realm of "pure" Math has no connections with the real world.

And when we say: "the logical realm of "pure" Math has no connections with the real world", don't we use philosophy?

As you can see Matt, I understand very well your game, and because I understand it I have the motivation to develop a better game which philosophy, our cognitive abilities and reality influence are legitimate parts of it.

 

[pig]

impressive use of enlarged letters, organic. the blue is also a nice touch.

good luck with developing a "better game". i am looking forward to a rigorous, logically consistent system which will be more efficient in describing real world problems than math is :)

 

[matt grime]

Organic, you also wanted me to show that I have understood you, well, in this thread we've pointed out that if you're so against proof by contradiction one can remove that part of the argument, but you've not commented upon that nicety.

Did you miss it?

Let X be the property "is a positive real number is smaller than any other positive real number"
Let Y be "the number is zero"

if d, a positive real number satisfies not(Y), then as d<d/2 is clearly not true, it follows that not(X) is true

ie not(Y)=>not(X), ie X=>Y

see, no assumption of mutually exclusive events.

Also I pointed out that given two statements, say A, B, that AandB false, which is what happens, is not meaningless, but tells you one of the two statements A and B is false.

DId you take time out to smell the roses and realize that any of those things had actually happened?

 

[Organic]

Invariant state 1) a not= b --> abs(a-b)=d > 0
Invariant state 2) d=d < e > 0

Therefore by (1) and (2) for any e > d, d > 0


Please show us the proof by contradiction, according what I wrote.



Also this time please show how x_n = limit-point in this particular curve:

http://phys23p.sl.psu.edu/~mrg3/mathanim/calc_I/Newtons.html

 

[matt grime]

What's an invariant state. And why would I wish to demonstrate something for that example that is probably not true, and is impossible to verify becuase it;s just a diagram, after all, if we look close enough, because of the limits of the diagram, that curve will have to be a straight line, possibly one pixel large long at the current distance. That curve doesn't have an equation describing it, you can't possibly know how it behaves if we 'zoom' in. How can I prove the impossible? I offered the observation that, in general, there is nothing to stop N-R iteration converging after a finite number of steps, I said nothing about that particular example, it was only your imprecise language that allowed that mistake to propogate.

 

[Organic]

Invariant state means: no matter how you zoom in, always you will find that:

1) a not= b --> abs(a-b)=d > 0
2) d=d < e > 0

Therefore by (1) and (2) for any e > d, d > 0

 

[matt grime]

but how on Earth does one 'zoom into' a proposition? how can i prove things using definitions that do not make sense to me. I have proved the assertion that if two real numbers, a and b, satisfy the property that for all e>0 |a-b|<e then a=b. I did it by contradiction, understood that you disbelieved it because you don't see how that method of proof works so I rewrote it without contradiction (ie using the contrapositive). I cannot help you learn how to understand proof by contradiction because it is part of your dogmatic and religious approach that you refuse to be open minded to it (you refusing to attempt to learn how it works). All of your reasons for not believing it are fanciful and illogical, and you can't teach someone to be logical, just show them how it works, and hope the use it. We've explained how it works, why don't you learn about it rather than automatically gain-saying it because someone like me, who you distrust automatically, told you about it? You cannot be hypocritical and hope not to lose respect.

 

[Organic]

who you distrust automatically

I am like a marsian that do not see infinity as you see it, and the way I see infinity your logical proposition does not hold.

The reason is this:

1) a not= b

Matt: By my way (1) is an hypothesis.

Organic: By my way (1) is an invariant state.

2) abs(a-b)=d < e > 0

Matt: a) By my way you compare d to a set S that includes in it all R members > 0 in this case d<d is impossible; therefore d must be = 0 --> a=b

Matt: b) Another version of my way is to say that e=d/2 but then |a-b|=d AND |a-b|<d/2 which is impossible; therefore a=b.

Organic: e and d relation remaining unchanged in any arbitrary scale that you choose, which means: d is always smaller then e but greater than 0, it means that e=d/2 is impossible because e > d/n > 0.

Organic: S is an open collection (has infinitely many elements) therefore cannot be completed by definition. Only finite collection can be a complete collection. Therefore there is no such thing like S which includes all r > 0.

Matt: e and d are fixed values.

Organic: e and d are variables, and both of them always greater 0.

 

[matt grime]

could you do everyone a favour? when you're about to switch and start talking about mathematics as only you consider it and not as the rest of the world understands it could you give us a signal? we prove things in context. if you're just going to artificially change the context and the rules at least have the decency to let us know so we can just file it under 'organic goes off on one again' for instance we state d is fixed (it is |a-b| and they are fixed), so if you're going to use another incompatible definition of it give us a heads up.

 

[Organic]

'organic goes off on one again'

I think that the one who goes on an off is you Matt, for example:

By your method, infinitely many elements have the same property like finitely many elements, and this is the reason that you have no problem to use words like 'all' and 'complete' together with infinitely many elements.

But sometimes you use the open interval Idea, and then no infinitely many elements can reach the limit.

Trough my point of view, a collection of infinitely many elements cannot be completed by definition, therefore I call it an open collection.

 

[matt grime]

Then you aren't using any of those words with the definition attached to them that other people give them. Therefore it is no wonder you are constantly saying things that are wrong on appearance, as everyone keeps telling you, you are using words with a different meaning from everyone else, and hte fact that what they say is 'wrong' if given your meaning to the words is not a particularly relevant issue because of your misuse of the terms.

 

[Organic]

Have you heard about the words "Paradigm Shift"?

Paradigm shift is like a mutation where mutation changes the system from within and not just adding another pretty thing to the existing system without changing any fundamental concept of it.

For example my organic numbers:

Let x be a general notation for a singleton.

When a finite collection of singletons have the same color, we mean that all singletons are identical, or have the maximum symmetry-degree.

When each singleton has its own unique color, we mean that each singleton in the finite collection is unique, or the collection has the minimum symmetry-degree.

Multiplication can be operated only among identical singletons, where addition is operated among unique singletons.

Each natural number is used as some given quantity, where in this given quantity we can order several different sets, that have the same quantity of singletons, but they are different by their symmetrical degrees.

In more formal way, within the same quantity we can define all possible degrees, which existing between a multiset and a "normal" set, where the complete multiset and the complete "normal" set are included too.

If we give an example of transformations between multisetes and "normal" sets, we can see that the internal structure of n+1 > 1 ordered forms, constructed by using all previous n >= 1 forms:

[b]1[/b]
(+1) = {x}

[b]2[/b]
(1*2)    = {x,x}
((+1)+1) = {{x},x}

[b]3[/b]
(1*3)        = {x,x,x}
((1*2)+1)    = {{x,x},x}
(((+1)+1)+1) = {{{x},x},x}

[b]4[/b]
(1*4)               = {x,x,x,x} <------------- Maximum symmetry-degree, 
((1*2)+1*2)         = {{x,x},x,x}              Minimum information's 
(((+1)+1)+1*2)      = {{{x},x},x,x}            clarity-degree
((1*2)+(1*2))       = {{x,x},{x,x}}            (no uniqueness) 
(((+1)+1)+(1*2))    = {{{x},x},{x,x}}
(((+1)+1)+((+1)+1)) = {{{x},x},{{x},x}}
((1*3)+1)           = {{x,x,x},x}
(((1*2)+1)+1)       = {{{x,x},x},x}
((((+1)+1)+1)+1)    = {{{{x},x},x},x} <------ Minimum symmetry-degree,
                                              Maximum information's  
                                              clarity-degree                                            
                                              (uniqueness)
[b]5[/b]
...

Can someone give me an address of some mathematical brach that researches these kind of relations between multisets and "normal" sets?

Thank you,

Organic

 

[matt grime]

Ah, paradigm shifts, automatic 40 point penalty on the crackpot index. Organic, I have forgotten more about this than you will ever learn, stop patronizing please.

 

[Organic]

automatic 40 point penalty on the crackpot index

1) patronizing? Who gives the points here, me?

2) Please show us a mathematical branch that define organic numbers as I show in my previous post.

3) Once you asked me to explain how N is not a complete collection by show n which is not in N.

My answer is very simple: natural numbers do not exists because of the existence of N, but because of the axioms that define them, N is only the name of the container that its content is infinitely many elements that can never be completed.

 

[matt grime]

How can we demonstrate organic numbers, as you've never defined them?

 

[Organic]

Here they are, in colors, to help you to understand their structures:

Let x be a general notation for a singleton.

When a finite collection of singletons have the same color, we mean that all singletons are identical, or have the maximum symmetry-degree.

When each singleton has its own unique color, we mean that each singleton in the finite collection is unique, or the collection has the minimum symmetry-degree.

Multiplication can be operated only among identical singletons, where addition is operated among unique singletons.

Each natural number is used as some given quantity, where in this given quantity we can order several different sets, that have the same quantity of singletons, but they are different by their symmetrical degrees.

In more formal way, within the same quantity we can define all possible degrees, which existing between a multiset and a "normal" set, where the complete multiset and the complete "normal" set are included too.

If we give an example of transformations between multisetes and "normal" sets, we can see that the internal structure of n+1 > 1 ordered forms, constructed by using all previous n >= 1 forms:

[b]1[/b]
(+1) = {x}

[b]2[/b]
(1*2)    = {x,x}
((+1)+1) = {{x},x}

[b]3[/b]
(1*3)        = {x,x,x}
((1*2)+1)    = {{x,x},x}
(((+1)+1)+1) = {{{x},x},x}

[b]4[/b]
(1*4)               = {x,x,x,x} <------------- Maximum symmetry-degree, 
((1*2)+1*2)         = {{x,x},x,x}              Minimum information's 
(((+1)+1)+1*2)      = {{{x},x},x,x}            clarity-degree
((1*2)+(1*2))       = {{x,x},{x,x}}            (no uniqueness) 
(((+1)+1)+(1*2))    = {{{x},x},{x,x}}
(((+1)+1)+((+1)+1)) = {{{x},x},{{x},x}}
((1*3)+1)           = {{x,x,x},x}
(((1*2)+1)+1)       = {{{x,x},x},x}
((((+1)+1)+1)+1)    = {{{{x},x},x},x} <------ Minimum symmetry-degree,
                                              Maximum information's  
                                              clarity-degree                                            
                                              (uniqueness)
[b]5[/b]
...

Can you give me an address of some mathematical brach that researches these kind of relations between multisets and "normal" sets?

Thank you,

Organic

 

[matt grime]

You're just defining certain types of partition functions. Why should we do your research? Do I patronize you? Tell you things you already know? Strange you don't seem to know lots of things. I'm certainly not polite to you, but that's because you don't earn respect from me, not that you'd want to obviously.

 

[Organic]

You're just defining certain types of partition functions

Have you seen before any use of these partition functions as I do?

 

[matt grime]

Yes and no. I have not seen people assign words to things as you do, but then as you never explain what any of those words mean that is highly irrelevant. You don't actually do anything with the things you write down as you admit yourself (and that scores you 50 more points on the crackpot index as well).

 

[Organic]

You don't actually do anything with the things you write down as you admit yourself

Please look here:

http://www.geocities.com/complementarytheory/CATpage.html

and show us an example of how I don't actually do anything with the things I write down.