The Foundations of a Non-Naive Mathematics

[matt grime]

doron, when you talk about "number" can we absolutely clarify that you mean the real numbers as we understand them in, say, calculus? Yes or No. because we are talking about these real numbers, and it is not clear what you are talking about since you've never offered an alternative explanation, hence our presumption you are using the real numbers as we know them already. You still appear to think that it is important what representative we use for a number, when it isn't: 1 and 2-1 are the same number, yet appear different, the same as 1/2 and 2/4. and if you're about to say something involving your new fangled terms, you must define what they are, and you must stop referring to things as the real numbers, when they aren't.

 

[Lama]

yet appear different,

Let us take your Idea and also say that you and I are the same person yet appear different, any number is actually zero yet appear different, and so on ,and so on...

By my system 'integral' and 'differential' are complementary properties, which simultaneously preventing/defining each other, and the result is infinitely many numbers that have internal complexity, which is based on symmetry/information complementary relations, that can be found on infinitely many different scales.

Multiplication and addition are their complementary internal operations.

The standard R members are based on one and only one building-block of my new system,
And its common property is: 0_redundancy_AND_0_uncertainty.

This building-block also standing in the basis of the excluded-middle reasoning.

Because of this limitation, a number is based only on the quantity concept, which through it 0.999… and 1 are look the same.

But when the minimal conditions for the existence of a number are both structural and quantitative, then 0.999… and 1 are different things.

 

[matt grime]

Oh dear, is that the best you can do? That makes it very easy to prove that what you are talking about aren't the real numbers, that's fine, but just don't say that they are [the real numbers] that's all. For instance, they could be thought of as representing some cauchy sequences, where they are different cauchy sequences, that doesn't stop them representing the same real number, in fact by definition, nothing more nor less, they must represent the same real number.

So, Doron, if it's a spade call it a spade, don't confuse objects like this, it only leads you to speak even more confusingly than you already do. And so all that you conclude applies only to your set of numbers not the real numbers. CF hurkyls post on your cardinal/organic number mistakes about 20 posts back.

You and I are different as human beings, yet if all that were required is a representative of a human being both of us are equivalent, in this arm wavy non-mathematical rubbish.

 

[Lama]

What you call real numbers are a shadow of my system.

Can you separate between something to its shadow?

 

[Lama]

CF hurkyls post on your cardinal/organic number mistakes about 20 posts back.

If you mean to these posts https://www.physicsforums.com/showpost.php?p=266256&postcount=77 https://www.physicsforums.com/showpost.php?p=267486&postcount=104 then since N,Q and R members are the shadows of my number system, I can talk about them by using my number system,
and this is exactly what I did here: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

 

[matt grime]

No, you can't, since in your system 0.999... and 1 are not equal, then you may not use them to talk about the real numbers where they are equal. Irrespective of your philosophical objections to that fact.

Just because S<T is a containment of sets does not imply that results about T mean anything about S.
A trivial example: In R there are no non-trivial ideals yet Z possesses many.

Incidentally, what do you mean by shadow? that's not a mathematical term, at least not in this context (it is in optimization). If you insist on using odd terms at least tell other people about them.

 

[hello3719]

A cute answer:

1/3 is not a fractal where 0.333... is a single path of a fractal.

They are not the same number exactly as 0.999... and 1 are not the same number.

When your logical system is based on an Included-Middle reasoning, the internal structural properteis of any mathematical element, cannot be ignored anymore.

is 1/2 still equal to 0.500... ?

 

[Lama]

Matt, think simple, if I say (and show) that standard system is the shadow of my system it means that I can explain your system by my system but you cannot explain my system by your system, because your system is no more than the quantitative shadow of my system.

 

[Lama]

is 1/2 still equal to 0.500... ?

If you take 000... as no information then 1/2=0.5

 

[hello3719]

so in your system the division operation will never yield a result with an infinite decimal expansion ?

What about pi ? how would you represent it ?

 

[Lama]

Well hello3719, there is no 'my system' here, because you can find the whole standard system as some particular case of this system.

For example, let us say that x,y system is a particular case of x,y,z system where z is ignored in this case.

We can do any x,y thing by an x,y,z system, but not vise versa.

My system can yields results with an infinite decimal expansions, for example please look at http://www.geocities.com/complementarytheory/9999.pdf pages 3,4.

Now for pi:

In the standard quantitative-only system, the internal information form of each number is ignored (by the analogy: we ignore z) ; therefore the fractal structure (that can be shown above in pages 3,4) of the decimal method is omitted and as a result, for example 1/3=0.33333...

But if any number is not less than quantitative/structural information form, we have the ability to distinguish between 1/3 and 0.333... by both quantitative and structural properties.

In short, we have more possibilities to do Math, and in the case of pi, each different base value define a new fractal of pi where the quantitative sum of each fractal is less then pi constant.


Also please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf including its entire links.

 

[terrabyte]

even in the current system, .333... does not accurately represent the number 1/3.

1/3 is a closed expression. .333... is open ended to and beyond infinity. to make it short, 1/3 is complete and rational, .333... is a continuing process to infinity and as such is irrational.

 

[hello3719]

even in the current system, .333... does not accurately represent the number 1/3.

1/3 is a closed expression. .333... is open ended to and beyond infinity. to make it short, 1/3 is complete and rational, .333... is a continuing process to infinity and as such is irrational.

you are not using the usual definition of "irrational", irrational merely means that it cannot be expressed as a ratio of 2 integers.

 

[zeronem]

even in the current system, .333... does not accurately represent the number 1/3.

1/3 is a closed expression. .333... is open ended to and beyond infinity. to make it short, 1/3 is complete and rational, .333... is a continuing process to infinity and as such is irrational.

A rational number is a number that can be expressed as the ratio of two integers.

A rational number can be expressed as either a fraction, or as a decimal number. If the decimal representation of a number goes on forever without repeating any pattern, then that number is an irrational number.

In this case, .33333... has a repeating pattern, and therefore is a rational number.

 

[hello3719]

But if any number is not less than quantitative/structural information form, we have the ability to distinguish between 1/3 and 0.333... by both quantitative and structural properties.

In short, we have more possibilities to do Math, and in the case of pi, each different base value define a new fractal of pi where the quantitative sum of each fractal is less then pi constant.

did you know that the "/" sign only represents the division operation ?

We can't talk about quantitive and structural properties when we compare an operation to its result.

Can you tell me how would you OBTAIN Pi ? You only showed how you would represent it not how you knew it was around 3.14.

 

[terrabyte]

you are not using the usual definition of "irrational", irrational merely means that it cannot be expressed as a ratio of 2 integers.

exactly. CANNOT be expressed. 1/3 is a complete expression. .333... is an incomplete expression. it cannot EVER be completed, to infinite digits or beyond.

the LIMIT of the expression .333... is 1/3 which means it will NEVER reach that number, no matter how many digits you extend it out to.

get it yet?

 

[hello3719]

exactly. CANNOT be expressed. 1/3 is a complete expression. .333... is an incomplete expression. it cannot EVER be completed, to infinite digits or beyond.

the LIMIT of the expression .333... is 1/3 which means it will NEVER reach that number, no matter how many digits you extend it out to.

get it yet?

LOL. Exactly .333... CAN BE EXPRESSED as a ratio of integers. Seems you have a flaw in your logic.

 

[Lama]

Pi is the ratio between circle's perimeter and circle's diameter.

We can take this ratio and represent it in infinitely many different quantitative/structural ways ,where '/' is not a division operation but a relation sign.

But no one of the representation is equal to pi constant.

 

[hello3719]

Pi is the ratio between circle's perimeter and circle's diameter.

We can take this ratio and represent in infinitely many different quantitative/structural ways ,where '/' is not a division operation but a relation sign.

What do you mean by '/' is a relation sign ? define relation sign

 

[hello3719]

Pi is the ratio between circle's perimeter and circle's diameter.

Please devise a method to find it using your more "complete system".

 

[Lama]

If you think that when I write quantitative/structural I mean quantitative divided by structural then to make it clearer I explained that '/' is used here to say that there is a mutual interactions between the two concepts.

 

[Lama]

Please devise a method to find it using your more "complete system".

1) No consistent system can be a complete system, and my system is incomplete.

2) Multi leveled parallel/serial Turing machine model, which uses any useful combination of my information building-blocks , instead of using only the binary buiding-block (which standing in the basis of the current Turing machine).

 

[hello3719]

If you think that when I write quantitative/structural I mean quantitative divided by structural then to make it clearer I explained that '/' is used here to say that there is a mutual interactions between the two concepts.

no, I meant when you write 1/3 we are talking about a division, and .33... is THE RESULT of the operation. This means that you can't compare the result with the operation.

 

[hello3719]

1) No consistent system can be a complete system, and my system is incomplete.

2) Multi leveled parallel/serial Turing machine model, which uses any useful combination of my information building-blocks , instead of using only the binary buiding-block (which standing in the basis of the current Turing machine).

I didn't say that it was complete, i said that you assumed before that our "actual" system is a mere shadow of yours , didn't you ?
If yes, then could you please show me in detailed steps how would you obtain Pi.

 

[Lama]

1/3 is not just an arithmetical operation of 1 divided by 3, but it is also the Q member 1/3.

The same we can say about pi (which is a R member), for example:

If the diameter is 1 then pi is pi/1.

Now you, can take 1/3 or pi and use the base value expansion method, which is a fractal way to represent numbers.

But because some fractals can be found in infinitely many different scale levels, then from the quantitative point of view they never fully represent 1/3 or pi.

 

[hello3719]

1/3 is not just an arithmetical operation of 1 divided by 3, but it is also the Q member 1/3.

The same we can say about pi, for example:

If the diameter is 1 then pi is pi/1.

well the Q set existence is based on the fact that the division operation is admitted in the system using the integers thus giving 1/3 a unique identity.
(saying "1/3 is not just an arithmetical operation of 1 divided by 3, but it is also the Q member 1/3." means nothing at all)

 

[hello3719]

The same we can say about pi, for example:

If the diameter is 1 then pi is pi/1.

That's the best you can do ?

 

[Lama]

Q are called rational numbers because these numbers are defined by the ratio that can be found between at least two integers, where the irrational numbers cannot be defined in this way, so as you see '/' is also used for a ratio.

Therefore (for example) 1/3 means the ration between 1 and 3, which is the rational number 1/3.

 

[hello3719]

But because some fractals can be found in infinitely many different scale levels, then from the quantitative point of view they never fully represent 1/3 or pi.

Who said that we want to use fractals to represent number theory concepts ?
The concepts we are talking about use some more basic defintions than the fractals' one.

 

[terrabyte]

LOL. Exactly .333... CAN BE EXPRESSED as a ratio of integers. Seems you have a flaw in your logic.

not so.

.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.

even if you have INFINITE digits with value 3, you STILL do not have the number that EQUALS 1/3. I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.

.333... is NOT 1/3

there is no flaw in my logic. just a failure in realization on your part.

 

[hello3719]

Q are called rational numbers because these numbers are defined by the ratio that can be found between at least two integers, where the irrational numbers cannot be defined in this way, so as you see '/' is also used for a ratio.

ratio and division is the exact same thing in mathematics, so why are you saying "also" ?

 

[Lama]

Who said that we want to use fractals to represent number theory concepts ?
The concepts we are talking about use some more basic defintions than the fractals' one.

If you use the base value expansion methods then you use a part of a fractal, whether you want it or not.

In this case you cannot ignore anymore the properties of a fractal.

 

[hello3719]

not so.

.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.

even if you have INFINITE digits with value 3, you STILL do not have the number that EQUALS 1/3. I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.

.333... is NOT 1/3

there is no flaw in my logic. just a failure in realization on your part.

OMG. Now this proves that you DON'T know what infinity means neither what rational and irrational means.

"the irrational computational result of dividing 1 by 3." this doesn't make sense at all. I will say it one more time, irrational simply AND ONLY characterizes a number THAT CAN'T be expressed as a ratio of 2 integers. It is a simple as this.

 

[terrabyte]

yes. and I've said it a few times as well. .333... IS NOT properly expressed by 1/3

step 1. take the number 6
step 2. Divide it by 3.

step 3. take the number 6
step 4. multiply it by .333... (i don't care how many digits you expand it out to)

compare results.

notice anything? :|

 

[Lama]

ratio and division is the exact same thing in mathematics, so why are saying "also" ?

Because 1/3 is also known as the constant symbol of the number that is produced by this division operation.

 

[hello3719]

If you use the base value expansion methods then you use a part of a fractal, whether you want it or not.

In this case you cannot ignore anymore the properties of a fractal.

refering to http://mathworld.wolfram.com/Fractal.html

"A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales"

Did you know that "scales" are consequences of the arithmetic operation of division AND NOT VICE-VERSA. This means that you can stop using fractals to explain your ideas and just use more basic concepts.

 

[hello3719]

yes. and I've said it a few times as well. .333... IS NOT properly expressed by 1/3

step 1. take the number 6
step 2. Divide it by 3.

step 3. take the number 6
step 4. multiply it by .333... (i don't care how many digits you expand it out to)

compare results.

notice anything? :|

k we have a problem of communication it seems. To clear things out,
When i write .333... i mean there is an INFINITY of 3's. so by definition of infinity i can conclude that 6 * .333... is equal to 2. So what's the problem ?

 

[Lama]

Did you know that "scales" are consequences of the arithmetic operation of division AND NOT VICE-VERSA. This means that you can stop using fractals to explain your ideas and just use more basic concepts.

Not correct, we can define scales also by mutiplication.

 

[hello3719]

Because 1/3 is also known as the constant symbol of the number that is produced by this division operation.

Logical flaws as always.
1/3 means in english " we are dividing 1 by 3"
so it doesn't have a double identity, like i said before don't compare the result with operation.

 

[zeronem]

not so.

.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.

Like I said again, an irrational number has no repeated pattern of integers in decimal notation of a ratio.

1/3 has a repeating pattern of 3's, which makes it rational. 1/7 has a repeating pattern of .142857... which makes it rational. However Pi, being the ratio of the circle's circumference to diamter, is an irrational number because it has no repeating pattern of integers in decimal notation.

\frac {1}{3} = .\overline {3}


\frac {1}{7} = .\overline {142857}

 

[hello3719]

Not correct, we can define scales also by mutiplication.

well sure we can, since division is just going backwards in the multiplication.
Still a scale ISN'T as basic of a definition as is division or multiplication nor , as a consequence, is fractal.
You didn't prove anything.

 

[Lama]

Logical flaws as always.
1/3 means in english " we are dividing 1 by 3"
so it doesn't have a double identity, like i said before don't compare the result with operation.

Let us make it simpler.

I choose @ as the result of 1/3 so 1/3 = @.

But If you use the base value expansion method, then you use a part of a fractal, whether you want it or not.

In this case you cannot ignore anymore the properties of a fractal.

And these properties can be cleary understood by:

-----------------------------------------------------------------

If we use a structural point of view in this case, then 0.9999... is a single path of a base 10 fractal ( http://www.geocities.com/complementarytheory/9999.pdf pages 3,4 ), that exists upon infinitely many scale levels that cannot reach 1.

Also we can say that 0.999... = 0.9+0.09+0.009+0.0009+... and we can clearly see that this infinitely long addition cannot reach 1.

Therefore 0.999... < 1.

Another example:

Please look at this beautiful Koch Fractal members.cox.net/fractalenc/fr6g6s.577m2.html[/URL]

Now let us say the there is a 1-1 map between each fractal level of 0.9999... to each different blue level of Koch Fractal.

0.9999... = 1 if and only if we cannot find anymore a 1-1 map between some 0.000...9 to some Koch Fractal blue level.

Since Koch Fractal can be found in infinitely many blue levels and each blue level has a 1-1 map with some 0.000...9 fractal level, then we can conclude that 0.999... < 1.

Also we can say that 0.999... = 1 if and only if the outer contour of this multi-leveled Koch Fractal can be a smooth curve with no sharp edges.

It is clear that the outer contour line is not a smooth contour in any arbitrary examined scale level.

Therefore 0.999... < 1.

From this model you also can understand what is a "leap".

In short, any transition between a non smooth curve to a smooth curve, cannot be done but by a phase transition leap that also can be described by a smooth_XOR_no-smooth connection.

This model is better than any "abstract" mathematical definition, which leads us to "prove" that 0.9999... = 1.

Also by this "proof" we simply ignore infinitely many information forms that can be found in 0.9999... fractal.

Now think how many information forms are ignored by this trivial and sterile approach of standard Math.

 

[Lama]

well sure we can, since division is just going backwards in the multiplication.
Still a scale ISN'T as basic of a definition as is division or multiplication nor , as a consequence, is fractal.
You didn't prove anything.

By these two operations we get the same result, which is a fractal.

Therefore the fractal property is the invariant and these operations are only tools to explore it.

To cealrly see and understand this fractal please look at http://www.geocities.com/complementarytheory/No-Naive-Math.pdf page 5.

 

[sparkster]

yes. and I've said it a few times as well. .333... IS NOT properly expressed by 1/3

step 1. take the number 6
step 2. Divide it by 3.

step 3. take the number 6
step 4. multiply it by .333... (i don't care how many digits you expand it out to)

compare results.

notice anything? :|

step 1. Take the number 1.
step 2. Divide it by 3.
step 3. Keep dividing
...
...
...
step 4,653. Keep dividing
...
...
...
step 3039209823752820. Keep dividing.

Notice anything?

 

[sparkster]

Your "rigorous" proof depends on excluded-middle black_XOR_white reasoning.

therefore you cannot deal with the complexity of 0.999... case.

Only Included-middle reasoning can deal with the complexity of this fractal.

In short, your "rigorous" proof is nothing but the image of your trivial black_XOR_white reasoning method.

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 littel world of non-math.

 

[sparkster]

A cute answer:

1/3 is not a fractal where 0.333... is a single path of a fractal.

They are not the same number exactly as 0.999... and 1 are not the same number.

When your logical system is based on an Included-Middle reasoning, the internal structural properteis of any mathematical element, cannot be ignored anymore.

You've already established that you can't do analysis. Surely you can do long division and see how inane this statement is.

 

[sparkster]

not so.

.333... is an incomplete expression. the irrational computational result of dividing 1 by 3.

So you're just going to use a unique definition of irrational number? That's fine, but don't expect anyone who actually does math to understand you (or take you seriously).

even if you have INFINITE digits with value 3, you STILL do not have the number that EQUALS 1/3. I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.

This is just silly. If there are an infinite number of 3's you can't add anymore to the end.

.333... is NOT 1/3

there is no flaw in my logic. just a failure in realization on your part.

You're right, there's no flaw in your logic, you just don't understand what irrational and rational numbers are.

 

[Hurkyl]

I can add 1 more digit to whatever your number is and come up with a number that is Closer or More Accurately Expressed as 1/3 than your number.

If there's a 3 in every position to the right of the decimal place, then where, pray tell, are you going to add one more digit?

 

[terrabyte]

i'm going to add 1 more digit wherever you "stop" at of course.

otherwise how do you expect to convey the "meaning" of your number to me?

1. we derive numerical meaning from the differences in number digits. 1 is clearly different from 2. 1.1 is slightly different from 1.2. the differences in these digits allow quantities to have meaning.
2. the quantity .333... has no meaning until it is brought into the realm of known quantities. hence it has to be definable within the boundaries of numbers within proximity to it AND be distiguishable as such. this may seem like a pithy statement with no meaning but hold on...
3. the number .333... for this number to be exactly 1/3 it must have infinite digits of 3. Infinite in the sense that they're unending, not in the sense that they're greater than all numbers.
4. define a number that is slightly less than this number. will .333...2 work? not really, that number is closer to 1/3 than .333... or maybe .333... where the number of digits 3 is (Infinity-1) <getting into cardinality with that but meh whatever>.
5. as you can see, because of the nature of digits stringing out to infinity it is impossible within the current system to define the number in relation to other numbers within proximity.

 

[terrabyte]

which is one more reason why it's irrational, not merely in the sense that it cannot truly be expressed as a ratio of two integers, but because the quantity is unplacable on the number line.