# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

Calculus has nothing whatsoever to do with infinitesimals.

The entire point of calculus (and its more formal uncle Real Analysis) is to eliminate consideration of infinitesimals, replacing them with rigorous definitions of the real numbers and limits.

Did you learn about $$\epsilon - \delta$$? No?
Now, of course, calculus includes a lot more than limits but with "Quack" one needs to do one small step at a time.

and therefore allow mathematics and science generally to cope with the infinite in a way that they could make sense of.

I am glad to see you two agree.
Now, seriously speaking, "Quack", we aren't talking about Newton, Godel and religion, we are talking about your basic misconceptions in terms of mathematics. So, going back to your original error, textbooks teach you that $$\frac{1}{\infty}$$ is indeed 0. <shrug>

An infinite universe interferes with my idea of an infinite God, sorry.
no need to apologize... You are at where you are at...

I am glad to see you two agree.
Now, seriously speaking, "Quack", we aren't talking about Newton, Godel and religion, we are talking about your basic misconceptions in terms of mathematics. So, going back to your original error, textbooks teach you that $$\frac{1}{\infty}$$ is indeed 0. <shrug>

Tach, we are not talking about what the text books say.. this is a philosophical discussion about mathematics.
I am well aware that the text books say in regards to 1/infinity = 0 [thanks to you] and all I am doing with my commentary is attempting to place that notion in the context of a well thought out but necessarily arbitrary and contrived system that mankind has evolved in dealing with the infinite.

Students should never forget what it is they are actually accepting when reading a text book. As an aside: The same applies to those reading a religious text... other wise referred to as "critical thinking" or "clear thinking" etc.

Tach, we are not talking about what the text books say..

But you should take a break from posting rubbish and FIND OUT what the books say.

But you should take a break from posting rubbish and FIND OUT what the books say.
Are you saying that Isaac Newton's work is rubbish?

You asked for me to support the notion that science already acknowledges that 1/infinity =/= 0

I posted this
The mere existence of calculus is my evidence.
Now show other wise or be considered as a fringe troll.
and we are still waiting for your well thought out and productive critique...
Care for another chocolate?

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Are you saying that Isaac Newton's work is rubbish?

Isaac Newton's rubbish was rubbish.

The notion that both infinity and the finite can co-exist could be exampled using the following diagram.. IMO

Each circle is a zero point with a diameter of 1/infinity
Each finite number is centered at the center of each zero point
so indeed 0.999... = 1 and so too does 1 = 1

Isaac Newton's rubbish was rubbish.
you clearly have not read the The Principles of Mathematics. I carefully read about 1/3, read another 1/3 without "checking" it line by line and the remaining 1/3 just skimmed. Knowing that in his era, only geometrical proofs were considered valid (like some today do not consider computer proofs "by exhaustion" of all possibilities in a computer, valid proofs) one who reads his work cannot help be impressed by how powerful geometric arguments are in the hands of an exceptional genius. I was so impressed that I read his optics too.

All you show by your statement is how ill-informed / ignorant you are.

If a curve comes infinitely closer to a line but never reaches it, then it would not be able to be at 0.999... on that axis...

If a curve comes infinitely closer to a line but never reaches it, then it would not be able to be at 0.999... on that axis...
I would think yes it would reach the line if the line was 1/infinity thick but never do so if the line had a zero thickness...
However this is not a standard description...

Yet, all textbooks clearly show $$\frac{1}{\infty}=0$$ (more generally, they show $$\frac{a}{\infty}=0$$), so you made up the above all by yourself.

Are you trying to show that what is printed on paper, bound, and called a "book" is truth? Does that concept also apply to the Bible? The Bible is a book and it says there is a God. Now what? Do you expect me to believe there is a God because it's printed in a book that there is a God??

$$\frac{1}{\infty}=0$$

...and then you go on to show me a magic trick where you start with 1 cookie and divide it into so many parts (of which you don't have a specific number to define just how many parts that is) that it vanishes into thin air! You can divide a cookie into so many parts that it disappears without a trace! Just like magic! Poof, now it's gone!! Now you see it, now you don't!

Bwahahahahahahaa

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I would think yes it would reach the line if the line was 1/infinity thick but never do so if the line had a zero thickness...
However this is not a standard description...
If 0.999... = 1, then the line couldn't ever approach at 0.999... or it would equal one, then it would no longer get infinitely closer, but actually be on the line! It wouldn't matter how thick it was, if it was approaching as 0.999... then it would be on that line. Remember, we are assuming that in all actuality, 0.999... = 1.

To me it seems like if you had the right curve, that it would disprove the postulate that says that curve gets infinitely closer but never reaches a line. It would be that or we would have to admit that 0.999... is not equal to one in order to keep that postulate for curves that approach lines.

If 0.999... = 1, then the line couldn't ever approach at 0.999... or it would equal one, then it would no longer get infinitely closer, but actually be on the line! It wouldn't matter how thick it was, if it was approaching as 0.999... then it would be on that line. Remember, we are assuming that in all actuality, 0.999... = 1.

To me it seems like if you had the right curve, that it would disprove the postulate that says that curve gets infinitely closer but never reaches a line. It would be that or we would have to admit that 0.999... is not equal to one in order to keep that postulate for curves that approach lines.
I think you have raised a valid question, however I am sure it has been met with an appropriate answer.
The test would be to see if the sloping line ever was able to become exactly parallel with the line, even if not touching the line. If it was indeed 0.999...= 1 then the line should become parallel I would think. But somehow I believe this would not happen...
A good test I think..
1:0.9
1:0.99
1:0.999
1:0.9999

and so on

Maybe someone else has a resolution?
I have a feeling that the limits function steps in and define the result based on the magnitude of the difference being less than any number
I am not sure how this impacts on the validity of 0.999...= 1 except as possibly an exception.

However if the line had a 1/infinity thickness to it then after an infinite number of digit steps it would resolve and the lines would touch and continue on as one. [I would speculate]

Originally Posted by Tach View Post
Yet, all textbooks clearly show (more generally, they show ), so you made up the above all by yourself.

In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, but not zero in size;
Note "in common speech..."

But I do wonder why have an infinitesimal at all if it equals zero why not just have a zero instead?
Why do you think that is Tach? [if you don't mind me asking]
I believe it has something to do with dimensionality. An infinitesimal is still able to be applied to 3 dimensional space where as zero is zero dimensional.

Note "in common speech..."

But I do wonder why have an infinitesimal at all if it equals zero why not just have a zero instead?

You will need to take an introductory class to calculus in order to find out. They explain the reason in class. When do you think you'll take a break from spamming the forum in order to better use your time learning?

you clearly have not read the The Principles of Mathematics. I carefully read about 1/3, read another 1/3 without "checking" it line by line and the remaining 1/3 just skimmed. Knowing that in his era, only geometrical proofs were considered valid (like some today do not consider computer proofs "by exhaustion" of all possibilities in a computer, valid proofs) one who reads his work cannot help be impressed by how powerful geometric arguments are in the hands of an exceptional genius. I was so impressed that I read his optics too.

All you show by your statement is how ill-informed / ignorant you are.

You got all that from my (perhaps not very funny) joke that "Isaac Newton's rubbish was rubbish?" Is there something wrong with your reading comprehension? Or are you off your autism meds today?

Did you learn about $$\epsilon - \delta$$? No?

I seem to remember covering that when I took calculus, decades ago. As I recall it was part of one way that limits are defined.

Now, of course, calculus includes a lot more than limits but with "Quack" one needs to do one small step at a time.

How do we get from the definition of limit to the assertion that 0.9999... = 1?

In other words, if we define a function as the successive addition of additional 9's to the end of 0.9, it would seem that the value of that function would approach 1 asymptotically. 1 would appear to be the limit of that function.

So what justifies the assertion that the value of the function ever equals the value of the limit?

(Layman has already addressed this, and it looks like the fundamental question here.)

Obviously we can define $$\epsilon$$, our margin of error so to speak, as small as we want, and the difference between the value of the limit and the value of the function will always be smaller than that arbitrary margin. Newton and Leibniz had already recognized essentially the same thing, which is why they felt justified in dropping infinitestimals from their calculations.

But what justifies our equating infinitely small with zero, not just as a calculating convienience but as a mathematical identity? That's a very different proposition and one that appears at least on its face to be false.

I seem to remember covering that when I took calculus, decades ago. As I recall it was part of one way that limits are defined.

How do we get from the definition of limit to the assertion that 0.9999... = 1?

In other words, if we define a function as the successive addition of additional 9's to the end of 0.9, it would seem that the value of that function would approach 1 asymptotically. 1 would appear to be the limit of that function.

So what justifies the assertion that the value of the function ever equals the value of the limit?

Because 0.(9) IS the very definition of the limit. Therefore , 0.(9) and 1 are the SAME. This has been a very long thread, I must have explained this multiple times.