# The Squared Circle

James;

No. You agreed the value is assigned to the limit L, and f(x) will become closer to L as x increases. It qualifies as convergent, but closer is not equal. The string/sequence of 1's remains a string of 1's. Multiply the string by 9 forming a string of 9's. It would be magic for the 9's to become 0's. If x in the f(x) approaches a specific value v within the interval, then it's possible for f(x) to equal the limit v.

No it isn't. The sample of the sequence is supposed to show the form of the terms, and typically an ellipsis (...) to indicate no last term. Using the convention of an index n implies the natural integers, or 'for all n'.
There are no infinite series, or infinite numbers, but there are processes (rules of formation) with the potential to form numbers of any size. An example is n'=n+1 which generates the set of integers. It can form greater numbers, but not a greatest number.

The common problem to these disputes is the abstract concept of the continuum with its unlimited division, and using 'infinity' as a number.
Chemistry and now quantum physics reveals the world as composed of discrete quantities of matter and discrete amounts of energy.
The last sentence is not relevant. This discussion is about pure mathematics, not natural science.

And for what it is worth, quantum theory and chemistry make extensive use of calculus and other mathematical concepts that rely on limits and converging infinite series. So to suggest they are some way ruled out by modern science is just bullshit.

It seems odd that phyti seems only to be considering real-world application, as if maths is only applicable to the real-world, rather than consider it in the abstract.

It seems odd that phyti seems only to be considering real-world application, as if maths is only applicable to the real-world, rather than consider it in the abstract.
Indeed. Next he’ll be telling us mathematics can’t deal with more than 3 dimensions, because that is all that physical space requires. There’s a basic failure to understand what pure mathematics is, by the look of it.

Sarkus;

Trying to get through to you certainly feels like it.But so what if there is nothing "infinite" in the human experience? Seriously. So what?Maths isn't about human experience. It isn't even necessarily about reality. It is an abstraction. It is filled with concepts that have no "reality" but are meaningful tools.

How can you describe something you haven't experienced, like an alien abduction?
Measurement was a necessary practice in early history. Tracking time, recording inventory, business transactions, building structures, etc. That is human experience.
Math as a language was developed over centuries of experience. Today it is the verification tool for science.

Yes, there is such a thing as "infinitely" long. Not in physical reality, but in the abstract world of maths there clearly are: the sequence of real numbers is infinitely long etc. Why are you only focussing on what is possible practically?

The set of integers is 'infinite', is a figure of speech for 'has no limit' or 'has no greatest element'. If it is without a boundary/limit it cannot be measured/counted. Boundaries are the requirements that enable measurement in a world of finite entities. Math as a tool is useful when its calculations match experience. I prefer to integrate a spherical volume and apply a density factor vs counting the number of particles of mass in the sphere.

yes you can. An infinite sequence can sum to a finite number, as we have been trying to get through to you. The sum of the sequence 1 + 1/2 + 1/4 + 1/8 + ... etc sums to a finite number

The algorithm for that series is to add 1/2 of the current term as the increment to the sum S. That increment also equals the difference |Limit-S|. Thus S never terminates.
That is what you inherit with the concept of continuous division (a continuum).
Being a geometric progression, the limit will be 2. You round up the current value which approximates the limit and claim it equals the limit of 2, because no one can imagine an infinite number. Anyone can say both a 1 meter stick and a 2 meter stick have an
'infinite' number of points, but that doesn't determine which one is longer.

Therefore they are the same number, for all numbers that are not equal have a different number half-way between them.

Except integers. There is no number between 4 and 5.
The sequence .9R has no last decimal position, (because it has been defined as such), so it never terminates.

I'm not making an abstraction real, and I am using infinity as a concept, not a number.

I mean using 'infinity' for measurement AS IF it's a number, which are abstractions.
Number is a concept. So are all the means of representation originating in the mind.
Mental images are real even if they only exist in the mind.

No I'm not. Maths requires no element of time. Humans do to perform mathematical functions, but maths itself requires no element of time. 2+2=4 requires no time to be correct, only for humans to do the sum. So no contradiction.

You stated:
"Maths doesn't require "time"."
"One can sum an infinite sequence in seconds."
Sure looks like it.

In the abstract world of maths, in this example there is a series of infinite oscillations, and they are completed in a finite time. Real world physics need not be applied. It is conceptual / abstract. It is maths.

Then it it's fiction.

Nope, I've given you all the information you need to come up with the answer I gave.There is no "equilibrium" in the maths I gave. Just an infinite series of bounces, all complete in a finite time. That's maths for you. Your insistence on only considering real-world applications is doing you no favours.
Then your math is for entertainment and is classified as 'recreational math'.

Sarkus;

How can you describe something you haven't experienced, like an alien abduction?
Measurement was a necessary practice in early history. Tracking time, recording inventory, business transactions, building structures, etc. That is human experience.
Math as a language was developed over centuries of experience. Today it is the verification tool for science.

The set of integers is 'infinite', is a figure of speech for 'has no limit' or 'has no greatest element'. If it is without a boundary/limit it cannot be measured/counted. Boundaries are the requirements that enable measurement in a world of finite entities. Math as a tool is useful when its calculations match experience. I prefer to integrate a spherical volume and apply a density factor vs counting the number of particles of mass in the sphere.

The algorithm for that series is to add 1/2 of the current term as the increment to the sum S. That increment also equals the difference |Limit-S|. Thus S never terminates.
That is what you inherit with the concept of continuous division (a continuum).
Being a geometric progression, the limit will be 2. You round up the current value which approximates the limit and claim it equals the limit of 2, because no one can imagine an infinite number. Anyone can say both a 1 meter stick and a 2 meter stick have an
'infinite' number of points, but that doesn't determine which one is longer.

Except integers. There is no number between 4 and 5.
The sequence .9R has no last decimal position, (because it has been defined as such), so it never terminates.

I mean using 'infinity' for measurement AS IF it's a number, which are abstractions.
Number is a concept. So are all the means of representation originating in the mind.
Mental images are real even if they only exist in the mind.

You stated:
"Maths doesn't require "time"."
"One can sum an infinite sequence in seconds."
Sure looks like it.

Then it it's fiction.

Then your math is for entertainment and is classified as 'recreational math'.
No, it is pure mathematics, that’s all.

phyti,

All we are saying is that when we divide 1 by 3 it produces the decimal 0.333R as the result. So that answer, with its non-terminating decimal, is equal to 1/3. You can prove this to yourself by doing the long division and dividing 1 by 3. You will see that it produces exactly the non-terminating decimal 0.333R and no other decimal but that one. So 1/3 = 0.333R in mathematics. Easy right?

You are making things far more complicated by trying to invent reasons why 0.333R might actually be less than 1/3. In the process of doing this, you reveal that you yourself are not using the non-terminating decimal 0.333R, but rather you are using a decimal that has a finite number of decimal places instead. You say things like humans don't experience infinity, and there is not enough time for us to write out the entire non-terminating decimal. These hand wavy statements just further reveal that you are not using 0.333R but something like 0.3333333333333333333333 instead. We would all agree with you that number is less than 1/3, but that number is not 0.333R. Do you see the difference?

No, it is pure mathematics, that’s all.
I guess if people want to remain ignorant then it's their prerogative. I'm done here, so I'll leave you to trying to get through. Good luck.

I note you still haven't told me what the number is between 0.999... and 1.
If A = 1 and B = 0.999... (or 0.9R) then what is (A+B)/2
For you to be able to prove that A and B are not equal, you must surely be able to show that there is a number that falls between 0.999... and 1.

If A = 1 and B = 2, there exists a number (A+B)/2 that falls between them. In this case 1.5.
So what is that number that lies between 1 and 0.999...?

Has phyti ever answered this? If not, it speaks volumes.

phyti:

No. You agreed the value is assigned to the limit L, and f(x) will become closer to L as x increases. It qualifies as convergent, but closer is not equal.
The series represented by 0.9999... converges to a sum of 1. Agreed?
The string/sequence of 1's remains a string of 1's.
An infinite string of 1s.
Multiply the string by 9 forming a string of 9's. It would be magic for the 9's to become 0's.
The point is that 0.999... = 1. The two notations refer to the same number, just as in base 3, 0.2222.... = 1.

If you disagree, then tell me what number falls between 0.999.... and 1. There must be at least one, if 0.999... < 1.
No it isn't. The sample of the sequence is supposed to show the form of the terms, and typically an ellipsis (...) to indicate no last term. Using the convention of an index n implies the natural integers, or 'for all n'.
Yes, and there's a countable infinity of natural integers.
There are no infinite series, or infinite numbers ...
I think Mr Cantor wants a word with you.
... but there are processes (rules of formation) with the potential to form numbers of any size. An example is n'=n+1 which generates the set of integers. It can form greater numbers, but not a greatest number.
I agree. In other words, the cardinality of the set of integers is an infinite number. Mr Cantor called it "Aleph zero".
The common problem to these disputes is the abstract concept of the continuum with its unlimited division, and using 'infinity' as a number.
There's a whole subfield of mathematics that deals with well-defined (i.e. not abstract) infinite numbers. There is even a "continuum hypothesis". Look it up!
Chemistry and now quantum physics reveals the world as composed of discrete quantities of matter and discrete amounts of energy.
I don't know why you imagine that chemistry and physics determine mathematics.

forum;

Here is a modern day Zeno scenario of the chase.
It is a product of imagination based on real world events.
The language of math works the same regardless of the source.

A person robs a bank close to an interstate highway. Police respond quickly, issue a description of the robbers car. A person with a scanner reports sighting the car 12 miles west on the interstate. If the robber's speed is 60 mph (the speed limit) to avoid getting stopped, and the police car pursues him at 90 mph, how long before they overtake him?

Using basic math with t in hrs.,
for the robber, distance r=12+60t.
for the police, distance p=90t.
If r=p then t(90-60)=12.
t=12/30=.4 hr or 24 min.
The distance is a finite 24 miles and the time is a finite .4 hr.

Where is anything that is 'infinite'?

Where is anything that is 'infinite'?
Same place as the price of eggs, probably.

Where is anything that is 'infinite'?
To drive 24 miles, one first has to drive 12 miles. Then another 6 miles. Then another 3 miles. Then another 1.5 miles. And so on. As a series we get:

24 = 12 + 6 + 3 + 1.5 + 0.75 + ...

which has an infinite number of terms. This is what Mr Zeno was discussing with his paradox of the arrow.

To drive 24 miles, one first has to drive 12 miles. Then another 6 miles. Then another 3 miles. Then another 1.5 miles. And so on. As a series we get:

24 = 12 + 6 + 3 + 1.5 + 0.75 + ...

which has an infinite number of terms. This is what Mr Zeno was discussing with his paradox of the arrow.
What was Zeno's point in the runner and the tortoise?

What was Zeno's point in the runner and the tortoise?

So, you don't see the point James made in post #212? Seriously?!?

All of your objections to 1/3=0.333... would also apply to the series James wrote, if they were valid.

There is not enough time to write out all of the terms, so you claim it is invalid, right?

It is an infinite series, and humans do not deal in infinities, so you claim it is invalid, correct?

So, you don't see the point James made in post #212? Seriously?!?

All of your objections to 1/3=0.333... would also apply to the series James wrote, if they were valid.

There is not enough time to write out all of the terms, so you claim it is invalid, right?

It is an infinite series, and humans do not deal in infinities, so you claim it is invalid, correct?

My example was to show the time and distance can be determined without using 'infinite' series.

Zeno was not promoting 'infinite'series, nor was he denying motion, since it was a common occurrence.
He was demonstrating the nonsensical results from assuming a continuum with its perpetual division.

This subject is another instance of 'truth will never be decided by an opinion poll'.

My example was to show the time and distance can be determined without using 'infinite' series.

Zeno was not promoting 'infinite'series, nor was he denying motion, since it was a common occurrence.
He was demonstrating the nonsensical results from assuming a continuum with its perpetual division.

This subject is another instance of 'truth will never be decided by an opinion poll'.

So, to you, it is a "nonsensical result" that the infinite series 12 + 6 + 3 + 1.5 + 0.75 + ... equals 24.

It is really not so difficult for the rest of us to understand. What decimal result do you get when you divide 1 by 3? Can you write it down? Can you say that it equals 1/3? The rest of us can do it, but you can not. So what advantage does your opinion offer?

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My example was to show the time and distance can be determined without using 'infinite' series.
Fine, but Zeno's point was different.
Zeno was not promoting 'infinite'series, nor was he denying motion, since it was a common occurrence.
He was demonstrating the nonsensical results from assuming a continuum with its perpetual division.
He thought that his reasoning was paradoxical. Hence the label "Zeno's paradox" is often attached to these kinds of examples.

I don't know how good the historical record is, for us to know what Zeno himself thought about any possible resolution of his paradox. But modern physics/mathematics certainly has a solution: the infinite series that sums to a finite number. Zeno recognised the infinite series in terms of distance travelled, but of course there's also a corresponding infinite series of times that it takes to travel the various distances. Divide one infinite series by the other, in effect, and we end up with a finite velocity. Moreover, it turns out that motion is possible after all, despite Zeno's "paradox". We can all breathe a sigh of relief and get on with our lives, comforted by the knowledge that infinite series solve the problem.