The Foundations of a Non-Naive Mathematics

[Hurkyl]

thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.

By definition, 1/3 is the solution to 3 * x = 1.

3 * .333... = .999... = 1

Thus .333... = 1/3.

 

[terrabyte]

Countably infinitly many...this is going in a cirlce, and I don't see your point.

my point is. it makes no sense to say "infinity is not a number" when it serves as a convenient <yet weightless> argument in a dialog yet proceed to USE it as a quantity in further discussions.

hypocrisy is sometimes deemed the worst of vices...

 

[hello3719]

rational is expressable EXACTLY as a ratio of two integers. in the case of 1/3 the computation is NOT exact, since it never ends up to and beyond infinity.

thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.

as it stands there IS NO rational decimal form for .333...

and trailing digits 0 such as .50000 <-- have no significance unless used as incremental spacers for digits with value such as .500001 <--

Could you please explain what "EXACT" means in mathematics ?
as i said before , "..." in 0.333... means that there is an infinity of 3's.
Do the division, it is a NON EVER ENDING division where we get a 3 in each step.

 

[sparkster]

rational is expressable EXACTLY as a ratio of two integers. in the case of 1/3 the computation is NOT exact, since it never ends up to and beyond infinity.

What do think a ratio is? 1/2 means 1 divided by 2 and 1/3 means 1 divided by 3. Why does the operation of division bother you? Why does division produce something "bad" when you have 1/3 and something "not bad" when you have 1/2? You seem to have a problem with infinity. Why? If you think there's something wrong with infinitly many digits, then the burden is upon you to demonstrate why.

thusly the form of expression .333... is not the direct translation of 1/3 from fractional (rational) into decimal (rational) form.

as it stands there IS NO rational decimal form for .333...

Can you give a formal proof?

and trailing digits 0 such as .50000 <-- have no significance unless used as incremental spacers for digits with value such as .500001 <--

Why not? What is special about the digit 0 there isn't there for 3?

Also, you ignored my other questions. I'll repeat them.

How much mathematical training have you actually had?

Clearly, 0.4 < 0.5 and 0.3983984 < 0.4 and .0000000000003 < .00000000000000003, right?

In the case of 0.333... and 0.5454... how is not clear that the latter is greater than the former?

 

[chroot]

terrabyte has been banned. He used to call himself ram1024, ram2048, ram4096, etc. We have banned this person three times already, yet he still does not seem to understand that he is not welcome here, and nor are his pointless threads.

If any of you see activity that you suspect is due to the same individual, please let the staff know so we can deal with it.

- Warren

 

[sparkster]

my point is. it makes no sense to say "infinity is not a number" when it serves as a convenient <yet weightless> argument in a dialog yet proceed to USE it as a quantity in further discussions.

When call something a number, we mean that it is a natural number, an/a irr/rational number, a complex number, a quaternion, etc. Infinity is none of these, but it's still a quantity. Why is this a problem?

hypocrisy is sometimes deemed the worst of vices...

Intersting that you accuse on hypocricy while you were the person complaining about "immature" behavior.

 

[Lama]

Can you exhibit a number between 0.333... and 1/3?

This is a very good question.

My answer is:

Because of the duality of any R member (which clearly can be shown here http://www.geocities.com/complementarytheory/No-Naive-Math.pdf in page 5) at least the entire infinite fractal representarions of 0.333... cen be found between fractal 0.333... and constant 1/3.

To make it cealer, if a=1 and b is any posivite R member < 1 and > 0 , then fractal b*0.333... can be found infinitely many times between a*0.333... and a*1/3.

By definition, 1/3 is the solution to 3 * x = 1

3 * .333... = .999... = 1

Thus .333... = 1/3.

If x is a fractal then your definition does not hold exactly as x/0 = 1 does not hold.

x holds only if x=1/3.

If you say that 1/3 is not a number but an operation between two numbers, then we can do this:

1/3 = @, therefore 3 * x = 1 iff x = @.

Form these examples we can learn (in my opinion) that there must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.

 

[Lama]

I used a proof by contradiction which you attempted (and failed) to do. I used "<" which you've tried to use again and again.

You reply with meaningless gibberish.

Will you actually address what is posted rather than ramble in you private little world of non-math.

1) the contradiction that you find and you use in your proof, depends on the logical reasoning that you use; therefore your proof is no more then a 'must-have' result of the basic laws of excluded-middle reasoning, which standing in the basis of your reasoning method.

In short, there is no one and only one universal law that leads us to find a one and only one possible result.

2) My logical reasoning is based on an included-middle reasoning, where the contradiction concept does not exist because two opposites are simultaneously preventing/defining their middle domain.

Therefore I cannot fail to produce a proof by contradiction in an included-middle reasoning framework.

The Included-middle reasoning framework and also its relation to an excluded-middle reasoning, is clearly and simply shown here: http://www.geocities.com/complementarytheory/CompLogic.pdf

3) If you read carefully http://www.geocities.com/complementarytheory/No-Naive-Math.pdf then I think that you will understand what is the meaning of ‘<’ or ‘>’ in my framework.

4) I think that you will not be able to understand my system, if you continue to use your basic aggressive attitude, which can be shown by the expressions that you use in your replies.

I am here for communication, not for war.


Something about understanding:

In my opinion, to understand something is to be simultaneously in and out of the framework of the explored thing.

It means that no-thing can be really understood only within its framework.

I think that this insight standing in the basis of any good scientific approach, because from one hand it gives us the motivation to find more general frameworks, and on the other hand we know that this is a 'never ending story' of “built-in” evolution process.

 

[hello3719]

Ok, then please write down all the axioms you are using.

 

[Lama]

It is too general, please be more specific, thank you.

Ok, then write down all the axioms you are using.

Why are you so aggresive? is it a hard thing for you to say 'please'?

 

[matt grime]

what on Earth does it mean to 'in the framework' of sometihng in order to understand it? i understand how a cd player works, but I'm not in any sense in a framework of a cd player... the mind boggles.

 

[Lama]

Direct experience baby, not just theory.

 

[matt grime]

oh, so how about theoretical physics such as black holes or string theory? why must a theory be experienced directly? well, obviously it mustn't, but that's your problem. of course if you must have direct experience then you can't possibly use the real numbers or any such since they can't be directly experienced (they don't exist, in any physical sense, and we all know how subjective you can get).

 

[Lama]

Matt,

Physics has two legs, the theoretical leg and the experimental leg, and they both complement each other.

Only by these two legs we can say that we understand something.

When we deal with ‘theory_only’ system then ‘in’ and ‘out’ frameworks are the inductive/deductive interactions of our ideas, or our local/global points of view.

 

[Lama]

Matt,

I thought about something and I'll be glad to know your opinion.

It goes like this:

When two violins in the same room are tuned with each other, if we play on one of them we find that the strings of the other violin are also vibrate.

Now let as say that intuition is our tuned instrument, and if a person expresses its intuitions by developing a way of thinking, the people that embrace this way of thinking probably share the same intuitions.

On the basis of these common intuitions a community can be established.

Let us say that this community is the first organization that deals with some part of the human knowledge, so in these early stages this community has no comparators on this part of the human knowledge.

Quickly this community becomes the most developed organization, which holds this part of the human knowledge, and other parts of human civilization look at this organization as the one and only one possible intuition which standing in the basis of a one and only one way (school) of thought (and I am not talking about variations, which are actually different brunches of the same way of thought, or the same school of thought if you like).

2500 years are passing and this school of thought survives because of two main reasons:

1) This way of thought was fitting to the needs of the human civilization along these 'slow' (linear) years.

2) Any other alternative intuitions (if they where at all) where put aside because:

a) They where not useful in their time.

b) And if they where useful and also a real alternative to the current school, then the current school used its power and money to block this alternative intuition by forcing its educational methods on the public.

We have to understand that intuitions cannot be learned, but a lot of external power can distort them until they lost their ability to be the source of a new school of thought.


The 120 century is the time where our civilization moved from linear time to non-linear time.

In this time the power of few holds the destiny of our civilization, and most of their power is based on the technical abilities that where developed by this school of thought, that was established 2500 years ago.

But our technical achievements, which are not balanced by another ways of thought, are like a government with no opposite.

We have learned that evolution needs diversity; otherwise we quickly get a dead planet.

The field of evolution in our non-linear time splits to "hardwhere" and "softwhere" parallel paths, where the hardwhere side is our technology and the softwhere side is our morality.

We can clearly see that there is no balance between the levels of these two paths, and this lack of balance in a non-linear time can quickly lead us to a dead-end street.

Therefore I think that we have to do the best we can to find the balance between our morality level and our technical abilities.

The first place that binds both paths is the language of mathematics.

In my opinion people how learn this powerful language, must first of all to develop their moral abilities by opening themselves to another intuitions which are not their intuitions and let them flourish in their communities.

By this way we develop our tolerance and learn how to live side by side, and if other intuitions are better then our intuition in this period of time, we do our best to help them flourish instead of trying our best to shut them down.

And we have the motivation to do that because we understand that we are all in the same boat.

My intuitions and ideas about the language of mathematics are different then the standard school of mathematics.

But in my opinion the most important difference, which I think fits to our non-linear time (more then the standard school) is that I include the mathematician cognition's ability to develop Math as a part of the mathematical research.

By this self-reference attitude I hope to develop the gateway that can connect between our moral abilities to our technical abilities.

And for that I need you help.

What do you think?

 

[sparkster]

It is too general, please be more specific, thank you.

It's not too general at all. I can list the field axioms and the least upper bound axiom, etc, that give tells how the real numbers work.

He's just asking you to list the axioms you're using. And if you do so, please try to it concisly and precisely.

 

[sparkster]

2) My logical reasoning is based on an included-middle reasoning, where the contradiction concept does not exist because two opposites are simultaneously preventing/defining each other.

Therefore I cannot fail to produce a proof by contradiction in an included-middle reasoning framework.

Don't you see how ridiculous this is? You say that contradictions do not exist in your "framework," but you cannot fail to prdocue a proof by contradiction. If contradictions don't exist, then a proof by contradiction cannot work! The proof by contradiction rests on A or not A being a tautology. If you don't hold to this, then you can't do proof by contradiction. You can post a "proof" that isnt' really one, and call it by "contradiction" which it wouldn't be, but don't use math terms unless you mean them. You'll just be harder to understand than you are usually.

3) If you read carefully http://www.geocities.com/complementarytheory/No-Naive-Math.pdf then I think that you will understand what is the meaning of ‘<’ or ‘>’ in my framework.

I think I expressed how arrogant and tiring it is for you to give out reading assignments. Are trying to get more hits on your website or something? Just post it for god's sake.

 

[arildno]

It's not too general at all. I can list the field axioms and the least upper bound axiom, etc, that give tells how the real numbers work.

He's just asking you to list the axioms you're using. And if you do so, please try to it concisly and precisely.

I think such a list will never come forth.
Usually, the inability to produce such a list, is indicative of the fact that the claimant is an arrogant ignoramus spouting incoherent blather; however we can't rule out completely the possibility that Lama does not belong in that august company.

 

[Lama]

Don't you see how ridiculous this is? You say that contradictions do not exist in your "framework," but you cannot fail to prdocue a proof by contradiction. If contradictions don't exist, then a proof by contradiction cannot work!

You missed the point because of your aggressive approach about my work.

Because contradiction does not exist in my framework, my ideas are not based on it at all, instead they are based on the complementary approach, where two opposites are simultaneously preventing/defining their middle domain.

Whole my work is based on this 'school of thought', where contradiction is the basis of your 'school of thought' of excluded-middle reasoning.

We are in two different worlds, and because of your aggressive attitude you still do not understand this, and still continue to examine my work from your 'school of thought' point of view.

So, the two simple questions that I want to ask you are:

1) Why are you so aggressive about my work?

2) Can you put aside your aggressive attitude before we continue?

 

[Lama]

He's just asking you to list the axioms you're using

Because my work examines several fundamental concepts of the language of mathematics, I need first to know what in what fundamental concept you are interested.

From your post I understand the you wish to see the list of the axioms that are related to the real numbers, so here they are:

A definition for a point:
A singleton set p that can be defined only by tautology ('='), where p has no internal parts.

A definition for an interval (segment):
A singleton set s that can be defined by tautology ('=') and ('<' or '>'), where s has no internal parts.

The axiom of independency:
p and s cannot be defined by each other.

The axiom of complementarity:
p and s are simultaneously preventing/defining their middle domain (please look at http://www.geocities.com/complementarytheory/CompLogic.pdf to understand the Included-Middle reasoning).

The axiom of minimal structure:
Any number which is not based on |{}|, is at least p_AND_s, where p_AND_s is at least Multiset_AND_Set.

The axiom of duality(*):
Any number is both some unique element of the collection of minimal structures, and a scale factor (which is determined by |{}| or s) of the entire collection.

The axiom of completeness:
A collection is complete if an only if both lowest and highest bounds are included in it and it has a finite quantity of scale levels.

The Axiom of the unreachable weak limit:
No input can be found by {} which stands for Emptiness.

The Axiom of the unreachable strong limit:
No input can be found by {__} which stands for Fullness.

The Axiom of potentiality:
p {.} is a potential Emptiness {}, where s {._.} is a potential Fullness {__}.

The Axiom of phase transition:
a) There is no Urelement between {} and {.}.
b) There is no Urelement between {.} and {._.}.
c) There is no Urelement between {._.} and {__}.

Urelement (http://mathworld.wolfram.com/Urelement.html).


The axiom of abstract/representation relations:
There must be a deep and precise connection between our abstract ideas and the ways that we choose to represent them.


(*) The Axiom of Duality is the deep basis of +,-,*,/ arithmetical operations.

Tautology means x is itself or x=x.

Singleton set is http://mathworld.wolfram.com/SingletonSet.html .

Multiset is http://mathworld.wolfram.com/Multiset.html .

Set is http://mathworld.wolfram.com/Set.html .

(By the way the diagrams in my papers are also proofs without words http://mathworld.wolfram.com/ProofwithoutWords.html )



The Axiom of the paradigm-shift:

Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current system.



Let us stop here to get your remarks.

 

[matt grime]

Here's a simple comment: you have not shown that the object which we call the real numbers satisfies any of those axioms. Nor do those axioms imply that the real numbers even exist (in a mathematical sense). Nor have you demonstrated that from these axioms alone can you construct anything, never mind something that is our real numbers.

 

[Lama]

Here's a simple comment: you have not shown that the object which we call the real numbers satisfies any of those axioms. Nor do those axioms imply that the real numbers even exist (in a mathematical sense). Nor have you demonstrated that from these axioms alone can you construct anything, never mind something that is our real numbers.

The real numbers are the shadow of my system.

You can still use Dedekind's sentence of the continuum that is based on the axiom of 'least upper bound', if you wish.

Before you continue Matt, please read https://www.physicsforums.com/showpost.php?p=270261&postcount=255

Thank you.

 

[matt grime]

You have not shown that there is any object that satisfies your axioms, and certainly the real numbers don't do it. you've failed to define almost all of the terms you use (the axiom of duality is, for instance, unintelligible).

so, start from the beginning, your "numbers" are a set S satisfying ... whatever your axioms are, prove that a model of this exists (you've not done that) then demonstrate what it is that you mean by "shadow" since it is not clear at all what that means. You might occasionally want to listen to a mathematician without being deaf to the objections: after all you want the dialogue.

 

[Lama]

what it is that you mean by "shadow"

You can still use Dedekind's sentence of the continuum (L < c < R) that is based on the axiom of 'least upper bound' ( http://mcraefamily.com/MathHelp/CalculusLimitUpperBound.htm ), if you wish.

the axiom of duality is, for instance, unintelligible

Did you read https://www.physicsforums.com/showpost.php?p=270261&postcount=255 ?

Also please look at http://www.geocities.com/complementarytheory/No-Naive-Math.pdf starting from page 5.

 

[matt grime]

the pdf does not define what any of those terms mean on page 5, I've read the other link to your post, not interested in commenting here, or there, and you've still not proven that there is a model for you axioms. all we know is that the real numbers aren't it, being only a shadow (something still not adequately defined), so what are they? that do you?

 

[Lama]

I wrote 'starting from page 5'.

Also I see that you chose to ignore my post to you at https://www.physicsforums.com/showpost.php?p=270261&postcount=255

So, let me put it this way:

There is no way to explain intuitions, it means that we already have them or not.

You do not share with me any common intuitions that are related to fundamental concepts of the Langauge of Mathematics, therefore you cannot understand my axioms.

On the contrary I can understand your axioms because they are based on intuitions that are less deep than my intuitions, and this is the reason why I am talking about a paradigm shift in the langage of Mathematics.

You will never understand me, and that is a clear fact after almost 2 years of dialogs between us.

So I want to ask you:

I'll be glad to continue our dialog, but why do you continue our dialog?

 

[matt grime]

does that reply imply that the terms on page 5 are defined on another page in the pdf? care to point out which one?

i understand that you're a crank, doron, and taking time out to answer yout trivial posts doesn't adversely affect my day, and after all it only takes for good men to do nothing for the crackpots to advance their theory, to paraphrase someone. better to debunk the garbage than let it fester.

 

[Lama]

I understand that you're a crank

So why do you continue your dialog with me if I am a crank, which his work cannot be understood by you?

better to debunk the garbage than let it fester.

How can you do that if you have no ability to understand even the most trivial thing of my work?

So, I am right about you.

You are no more then a full_time_job_sanitarian of the current school of thought of the Langauge of Mathematics.

 

[Hurkyl]

Let me point out one problem. (More later)

The Axiom of the weak limit:
No input can be found by {} which stands for Emptiness.

The Axiom of the strong limit:
No input can be found by {__} which stands for Fullness.

These are the only axioms that involve the terms "no input", "found", "{}", and "{__}". Thus, the only theorems you can prove about these three terms must be proven with only these two axioms.

Some examples of why this is a problem:
You cannot prove {} and {__} are different.
You cannot prove anything can be found by anything.
You cannot prove the existence of any input.

 

[Lama]

1) axioms are not proven

2) |{}|=0 , |{_}|=1 (The 1 continuum)

Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (please start from page 10 until the end of my article, if you wish to understand my work).