The Foundations of a Non-Naive Mathematics

[sparkster]

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.

You really don't understand math do you? That's just sad.

 

[Hurkyl]

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

Stop what?

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.

There is no (right) end to .333...

 

[terrabyte]

thus it will NEVER equal 1/3

 

[Hurkyl]

thus it will NEVER equal 1/3

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

 

[terrabyte]

.333... is not a rational number.

can you exhibit a number slightly greater than or less than .333... ?

 

[Hurkyl]

Yes, lots.
0.3 < 0.333... < 0.4
0.33 < 0.333... < 0.34
0.333 < 0.333... < 0.334
0.3333 < 0.333... < 0.3334
...

Pick any upper bound for "slightly", and I can find one of these inequalities such that the smaller and larger are slightly different than 0.333...

What is the point of this exercise?

 

[terrabyte]

k expand those out to infinity. infinite precision, so-to-speak

all of those numbers cease to have distinction

 

[terrabyte]

.333...2 and .333... and .333...4 are all the same number.

there is no designation or incrementation to distiguish their properties of being different numbers because of the current definition limitations of "Infinity"

that's the purpose of the exercise

 

[sparkster]

.333...2 and .333... and .333...4 are all the same number.

there is no designation or incrementation to distiguish their properties of being different numbers because of the current definition limitations of "Infinity"

that's the purpose of the exercise

.333...2 and .333...4 aren't numbers. They're products of your misunderstandings. If "..." is means inifinitly many, you can't have infinitely many 3's and then add a 2 or a 4.

 

[Hurkyl]

.333...2 and .333... and .333...4 are all the same number.

No... one of those is a number; the other two are gibberish.


Anyways, you've evaded my response. None of the approximations I intend to make have an infinite number of nonzero terms, however if you pick any positive value as an allowable tolerance for error, one of my approximations will be within this tolerance.


It seems you are trying to ask me to tell you what the "next" decimal number is, but there is no such thing. (If there was a "next" number, then what happens if I take their midpoint?)


And I'll ask again, what does all of this have to do with whether 1/3 = 0.333...?

 

[terrabyte]

then you can't define a number "slightly" greater than or "slightly" less than that number, and it has no meaning.

hence irrational

 

[Hurkyl]

then you can't define a number "slightly" greater than or "slightly" less than that number, and it has no meaning.

You said that 1.1 is slightly different than 1.2.

0.333 is quite a bit closer to 0.333... than 1.1 is to 1.2.

Were you wrong earlier when you said 1.1 is slightly different than 1.2?


And I'll ask again, what does this have to do with 0.333... = 1/3?

 

[zeronem]

With your logic terrabyte, you would equally assert that 1/7 = .142857... is a irrational number. Once again this proves your mis-understanding in Math.

You fail to realize that some rational numbers particularly 1/3 and 1/7 in there decimal notation run through a pattern of integers continuously. However irrational numbers have no significant pattern in there decimal notation.

1/3 = .333333333333333333333333333333333333333333333333333333333333333333

1/7 = .142857142857142857142857142857142857142857142857142857142857142857

Pi = 3.141592654,

There is no pattern in Pi, therefore it is irrational and such a number cannot be described as a ratio of two integers. We have only found fractions that closely resemble Pi that our rational. like 22/7 = 3 + 1/7

 

[terrabyte]

No... one of those is a number; the other two are gibberish.

Anyways, you've evaded my response. None of the approximations I intend to make have an infinite number of nonzero terms, however if you pick any positive value as an allowable tolerance for error, one of my approximations will be within this tolerance.

It seems you are trying to ask me to tell you what the "next" decimal number is, but there is no such thing. (If there was a "next" number, then what happens if I take their midpoint?)

And I'll ask again, what does all of this have to do with whether 1/3 = 0.333...?

the distinction or uniqueness of numbers has infinite precision, meaning no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.

that distinction is obliterated by using an infinite string of digits to express a quantity.

case in point our little exercise to determine the closest numbers to .333... the exercize would have worked for any number, i was just using it to illustrate that when we take digits and extend them out to an arbitrar "infinity" we lose the distinction that determines WHAT numbers actually are.

without incrementation, the ability to add MORE digits to further define value, you hit a logical dead end.

this is completely due to the structuring of infinity

you ask how this has to do with .333... and 1/3 well it goes way beyond those two numbers and their properties and into the intrinsic values and quantities of ALL numbers, and the limitations of our mathematical and numerical system.

 

[Hurkyl]

the distinction or uniqueness of numbers has infinite precision, meaning no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.

that distinction is obliterated by using an infinite string of digits to express a quantity.

Would you care to prove your point? Give me two decimals representing different quantities, and I'll find a decimal representing a number between them. (One way is I could simply add them and divide by 2)

 

[terrabyte]

that's the point.

when expanded out to infinite digits you can NOT create different quantities. there is no longer any room for incrementation.

we just did the exercise expanding .32 <.333... <.34 | .332 < .333... < .334

you ran into the same roadblock

 

[hello3719]

how old are you terrabyte ? It seems you don't understand anything.

 

[hello3719]

.333...2 and .333... and .333...4 are all the same number.

there is no designation or incrementation to distiguish their properties of being different numbers because of the current definition limitations of "Infinity"

that's the purpose of the exercise

Still don't understand the definition of infinity, very sad.

 

[terrabyte]

how old are you to make comments that have no basis on what's being debated?

where is your facts to back up this "argument" that i don't understand anything?

 

[Hurkyl]

there is no longer any room for incrementation.

Which is fine, because the real numbers don't have incrementation.

You claimed:

the distinction or uniqueness of numbers has infinite precision, meaning no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.

that distinction is obliterated by using an infinite string of digits to express a quantity.

I ask you to present two decimals representing different quantities for which I cannot find a decimal between them.

 

[terrabyte]

instead of hollow "very sad" remarks how about you explain what YOU understand infinity to be and how it fits into the picture.

if you have nothing relevant to say, by all means, keep quiet

 

[hello3719]

how old are you to make comments that have no basis on what's being debated?

where is your facts to back up this "argument" that i don't understand anything?

You can't even accept the definition of irrational neither infinity. Second of all mathematics isn't a question of "debate". This is not a politics forum.

 

[chroot]

We sure seem to have a lot of arrogant, ignorant kids on the forum all of a sudden. Are you all friends? Why are you here?

- Warren

 

[hello3719]

you think that .333...4 is a number where there's an infinite of 3's. Tell me how can you add a 4 if there is a NON-ENDING quantity of 3's?

 

[terrabyte]

real numbers DO have incrementation AND distinction.

every digit 0-9 that is different from one in the same location of another number DEFINES difference.

this is intrinsic AND integral to the structure and utilization of the number system.

when you put a limit on the ability of that system to function of course discrepancies will arise. which is exactly what "Infinity" is. a Limit.

don't come back and say it ISN'T, hello3759 (i know that's going to be your next pithy argument) because it most certainly IS. if it isn't define me a number greater than it.

there isn't one is there?

 

[Hurkyl]

real numbers DO have incrementation

On what grounds do you suggest that? Remember your own statement:

no matter how close you come to a number you can ALWAYS cut the distance to reveal another number closer.

If there was incrementation, then what comes between a number and its increment?

 

[hello3719]

don't come back and say it ISN'T, hello3759 (i know that's going to be your next pithy argument) because it most certainly IS. if it isn't define me a number greater than it.

there isn't one is there?

Greater than .333... ? no problem
why not 0.4?

 

[terrabyte]

a smaller increment.

always

 

[terrabyte]

greater than infinity

 

[hello3719]

greater than infinity

infinity isn't a number

 

[terrabyte]

semantics

say something that has meaning, Hello3719, PLEASE

 

[hello3719]

why doesn't it has meaning ?
it's true it isn't a number unless you can prove it

 

[terrabyte]

fine, you stick to that then it's absolutely impossible for you to utilize it as a number.

those are your guidelines.

now express the quantity of how many digits .333... has with values of 3?

 

[sparkster]

terrabyte, how much formal mathematics have you actually had? You seem to be a bright, but ignorant, person.

when expanded out to infinite digits you can NOT create different quantities. there is no longer any room for incrementation.

Could you explain this?
Clearly, 0.4 < 0.5 and 0.3983984 < 0.4 and .0000000000003 < 0.00000000000000003, right?

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

If this isn't what you meant, could you elaborate?

Also, is 1/2 = .5 = .5000...?

How do you technically define rational and irrational numbers?

 

[sparkster]

fine, you stick to that then it's absolutely impossible for you to utilize it as a number.

those are your guidelines.

now express the quantity of how many digits .333... has with values of 3?

It has as many digits as there are natural numbers.

 

[terrabyte]

It has as many digits as there are natural numbers.

fine, how many natural numbers are there then?

you're just digging a deeper hole :D

 

[hello3719]

fine, you stick to that then it's absolutely impossible for you to utilize it as a number.

those are your guidelines.

now express the quantity of how many digits .333... has with values of 3?

by notation i am using " ... " to mean never ending quantity of 3's.
why would you say that "never ending" is a number ?
Keep focus.

 

[sparkster]

fine, how many natural numbers are there then?

you're just digging a deeper hole :D

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

 

[Hurkyl]

how many natural numbers are there then?

The natural numbers have cardinality \aleph_0 and order type \omega.

 

[terrabyte]

terrabyte, how much formal mathematics have you actually had? You seem to be a bright, but ignorant, person.


Could you explain this?
Clearly, 0.4 < 0.5 and 0.3983984 < 0.4 and .0000000000003 < 0.00000000000000003, right?

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

If this isn't what you meant, could you elaborate?

Also, is 1/2 = .5 = .5000...?

How do you technically define rational and irrational numbers?

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 <--

 

[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'?