Refuting the counters to Zeno's paradox

[The Dagda]

Secondly, reasoning was provided for challenging the basis for a particlar axiom. That reasoning was recently re-posted, about 2/3 down page 13. It's there for you to consider, if you so choose. It amounts to:

That is: (1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 +....) is only true for definite/finite points of this series.

I think the problem here is that you assume the equality applies to the finite sum, rather than the infinite series. If that were true you might have a point, but the equality applies to the infinite series itself. It is an infinite number of finite equalities:

1/2 = 1/2 & 1/4 = 1/4 & 1/8 = 1/8 ...

Or simply 1/2n = 1/2n
For all n = 1...

This is an absolutely foundational axiom of mathematics and logic.
Perhaps you want to refute the logical AND?

This does not rely on the infinite series having a finite sum, so the equality and substitution are valid.

At least he's getting a free maths lesson eh.

[jamest]

Infinitessimals aren't real any more than infinities hence the limit is what is really happening and is asymptotic or 0 at lim infinity.

When you say that infinitessimals aren't real, are you implying that the world is constituted of discrete elements and that there is a smallest value for each of these elements?

According to empiricism/materialism the smallest probable entity is a quanta.

The smallest possible concept is mathematical and cannot be proven by empiricism.

the mass of photons for example has a lower limit of ~1x10^-60kg

I need to be clear on what you are saying, for reasons that will become apparent. Are you saying that there ARE fundamental particles which are not reducible to anything else? And are you saying that there are fundamental units of space/time/spacetime that are also not reducible to anything smaller? It certainly seems so, but I'd rather have you confirm this, unambiguously, before I proceed.

[jamest]

Secondly, reasoning was provided for challenging the basis for a particlar axiom. That reasoning was recently re-posted, about 2/3 down page 13. It's there for you to consider, if you so choose. It amounts to:

That is: (1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 +....) is only true for definite/finite points of this series.

I think the problem here is that you assume the equality applies to the finite sum, rather than the infinite series.

No, I reasoned that any equality must apply to finite series of terms.

If that were true you might have a point, but the equality applies to the infinite series itself. It is an infinite number of finite equalities:

1/2 = 1/2 & 1/4 = 1/4 & 1/8 = 1/8 ...

Or simply 1/2n = 1/2n
For all n = 1...

This is an absolutely foundational axiom of mathematics and logic.
Perhaps you want to refute the logical AND?

This does not rely on the infinite series having a finite sum, so the equality and substitution are valid.

Go back to the post where I discuss the equality of unending lines. There's no point in me repeating what I've already said.

[GrahamH]

Secondly, reasoning was provided for challenging the basis for a particlar axiom. That reasoning was recently re-posted, about 2/3 down page 13. It's there for you to consider, if you so choose. It amounts to:

That is: (1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 +....) is only true for definite/finite points of this series.

I think the problem here is that you assume the equality applies to the finite sum, rather than the infinite series.

No, I reasoned that any equality must apply to finite series of terms.

If that were true you might have a point, but the equality applies to the infinite series itself. It is an infinite number of finite equalities:

1/2 = 1/2 & 1/4 = 1/4 & 1/8 = 1/8 ...

Or simply 1/2n = 1/2n
For all n = 1...

This is an absolutely foundational axiom of mathematics and logic.
Perhaps you want to refute the logical AND?

This does not rely on the infinite series having a finite sum, so the equality and substitution are valid.

Go back to the post where I discuss the equality of unending lines. There's no point in me repeating what I've already said.

There are no 'unending lines' in the equality.

Well, I have only addressed Xamonas Chegwé's mathematical proof. So, anything I have said only applies to the axioms of that proof, though perhaps other mathematical proofs suffer from the same problem... I dunno.
Secondly, reasoning was provided for challenging the basis for a particlar axiom. That reasoning was recently re-posted, about 2/3 down page 13. It's there for you to consider, if you so choose. It amounts to:

That is: (1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 +....) is only true for definite/finite points of this series.

You have not shown any flaw in the mathematical proof.

In your OP:

... S = the sum of the series. But 1/2 + 1/4 + 1/8 + 1/16 + ...., is the series itself, not summed. So, what the author of this math has done, is that he's assumed that the series has a sum, so that he can say "But, if we discard the first term, this is just S.".

The whole point of the math is to try and prove that the series can be summed - not just to assume that it has a sum and then use that assumption to prove the sum of that series.

The proof does not assume the series has a sum. The equality is the series itself, infinite as defined, not an assumed finite sum.

Zeno got the maths WRONG, hence there is no paradox.
You got the maths WRONG, hence there is still no paradox.

[jamest]

Go back to the post where I discuss the equality of unending lines. There's no point in me repeating what I've already said.

There are no 'unending lines' in the equality.

Of course not, but the principles are the same: unending lines are never equal except in definite terms of finiteness or the places that they move through. Likewise, infinite series of numbers cannot be equal except in their finiteness and the numbers that are used.

Well, I have only addressed Xamonas Chegwé's mathematical proof. So, anything I have said only applies to the axioms of that proof, though perhaps other mathematical proofs suffer from the same problem... I dunno.
Secondly, reasoning was provided for challenging the basis for a particlar axiom. That reasoning was recently re-posted, about 2/3 down page 13. It's there for you to consider, if you so choose. It amounts to:

That is: (1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 +....) is only true for definite/finite points of this series.

You have not shown any flaw in the mathematical proof.

The flaw is in the rational basis for utilising the axiom which led to the mathematical proof.

... S = the sum of the series. But 1/2 + 1/4 + 1/8 + 1/16 + ...., is the series itself, not summed. So, what the author of this math has done, is that he's assumed that the series has a sum, so that he can say "But, if we discard the first term, this is just S.".

... The whole point of the math is to try and prove that the series can be summed - not just to assume that it has a sum and then use that assumption to prove the sum of that series.

The proof does not assume the series has a sum. The equality is the series itself, infinite as defined, not an assumed finite sum.

Yes I know this. You're going back several pages to soon after XC presented his revised proof (without 'S') and I just misunderstood him. That's why I changed tact and moved onto the present route of counter.

Zeno got the maths WRONG, hence there is no paradox.
You got the maths WRONG, hence there is still no paradox.

I haven't done any math. All I have done is to present a rational challenge to the utilisation of an axiom that led to some math. You don't appear to have a clear understanding of the course of this discussion.

[GrahamH]

OK, your repeat of the criticsm of the equality may have misled me to think you still didn't get the equivalence. Let's see what I take to be your latest formulation...

I'd like to add that it was never my initial intention to focus upon mathematical proofs that claim to have countered Zeno's reasoning - I just became aware of rational flaws inherent within the premises of the math that XC has employed and decided to deal with this first. Therefore, my post from last night (UK time), still has relevance... and nobody has addressed that at all.

With my last post in mind - which attempted to check the divergence from this original theme - I shall now bump the contents of last night's post, in the hope that somebody will address it:

I was very careful never to perform a single operation on the 'sum' as a whole. Every step involved actions on individual terms. That was the whole point in rewriting the proof (at great length, I might add.)

Firstly, I sincerely thank you for the effort and time that you have given to this thread.
Secondly, if I - as is apparent - misunderstood the basis of the second proof, then I apologise.
Thirdly, please read my last post to Jerome - any attempted counter here, by me, is reducible to a philosophical consideration of any math that have been forthcoming. I am not attempting to 'correct your math' per se.
Finally, I now actually do understand the basis of your 2nd proof and will proceed from that...

My counter to your 2nd proof will still be upon your initial foundation, though. So, you start with:

[pre]1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 + ....[/pre]

There is still a problem here, even if we are not considering 'the sum' of anything. The problem now has to focus upon the ellipsis (... ), as SD mentioned.

... Here, you are equating one series with another. But, the problem is this: if the series of numbers (1/2 + 1/4 + 1/8 + 1/16 + ....) has no end (is infinite), then any purported equivalences of that set must be finite, by logical default. That is, there can be no equivalences of anything unless it is in a definite/finite state. This is the my rational conclusion as per why you cannot state that:

1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 + ....

Thus, I am trying to argue that nothing is equivalent to anything else (including itself), unless that thing is in a finite state of being. Of course, I recognise that this requires further explanation, so I shall proceed:

Is the equivalence of a line without-end, just a line without-end? No, since both lines could be running through different places, at different times.

If the lines are defined identically then how can they differ?

The series 1/2 + 1/4 + ... is defined precisely to infinity. The formulation of every term is precise and in each case the same formulation applies.
Every term, to infinity, can be paired and is equal.

The lines, if defined in identical terms, must be equal.

Such unending lines cannot be equivalent, then. So, what basis is there for equating unending lines? Not in their unendingness - as has been explained - but in their 'endedness'. That is, no unending line can be equivalent to another except in finite/definite terms. That is, the equivalence of one thing to another, demands the utilisation of definite/finite facts to facilitate that equivalence. Therefore, there is no equivalence of an unending line - even with itself - unless definite/finite claims impose an equivalence on such an unending line.

That sounds like bullshit to me. The equivalence is precise in form, defined and infinite.

An unending line defined by y - 2x + 3 = 0 is exactly equal to itself without needing any actual numbers to be calculated.
The identical line defined by y - 2x + 3 = 0 is indeed identical, for all x,y to infinity.

That is: (1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 +....) is only true for definite/finite points of this series. The point being that since 'infinity' is an unknown quantity - and is neither definite nor finite - that the utilisation of the ellipsis (... ) means that there is no equivalence to anything beyond that which is definite or finite. I.e., one cannot equate anything with an ellipsis with anything else with an ellipsis, including 'itself'.

This could make sense only if the ellipsis did not refer to precisely defined terms, but the terms are precisely defined as half the previous term, all the way to infinity. The never becomes uncertain, even when actual numbers become unmanageable. You are attempting to refute A=A.

I.e.: 1/2 + 1/4 + 1/8 + 1/16 + .... ≠ 1/2 + 1/4 + 1/8 + 1/16 +.... [except at definite/finite points of the series].

Of course, if correct, this counter renders any subsequent math as null & void.

Perhaps this is difficult to understand... I dunno. But, just ask and I will try to elucidate further. It beats responding with posts of near-infinite boobs, anyway. (edited to add that this is a generalised statement).

Again, with my previous post in mind, what I said here does still have relevance, and therefore requires attention.
Cheers.

I think it has had more attention than it deserves.

[Surendra Darathy]

Such unending lines cannot be equivalent, then. So, what basis is there for equating unending lines? Not in their unendingness - as has been explained - but in their 'endedness'. That is, no unending line can be equivalent to another except in finite/definite terms. That is, the equivalence of one thing to another, demands the utilisation of definite/finite facts to facilitate that equivalence. Therefore, there is no equivalence of an unending line - even with itself - unless definite/finite claims impose an equivalence on such an unending line.

That sounds like bullshit to me. The equivalence is precise in form, defined and infinite.

An unending line defined by y - 2x + 3 = 0 is exactly equal to itself without needing any actual numbers to be calculated.
The identical line defined by y - 2x + 3 = 0 is indeed identical, for all x,y to infinity.

I.e.: 1/2 + 1/4 + 1/8 + 1/16 + .... ≠ 1/2 + 1/4 + 1/8 + 1/16 +.... [except at definite/finite points of the series].

Of course, if correct, this counter renders any subsequent math as null & void.

Again, with my previous post in mind, what I said here does still have relevance, and therefore requires attention.
Cheers.

I think it has had more attention than it deserves.

Not only that. Someone who can do math will see that the line y - 2x + 3 = 0 is identical to the line 2x - y - 3 = 0. That two such different-looking expressions exactly describing an unending line should be identical in all respects to the unending line with which they are identified is only a puzzle to someone puzzled by the concept of identity, z = z.

For someone to toss out the identity axiom in order to try to establish a new metaphysics is a radical step, and we can see where it has led in this case.

More ways to go wrong.

[jamest]

That is: (1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 +....) is only true for definite/finite points of this series. The point being that since 'infinity' is an unknown quantity - and is neither definite nor finite - that the utilisation of the ellipsis (... ) means that there is no equivalence to anything beyond that which is definite or finite. I.e., one cannot equate anything with an ellipsis with anything else with an ellipsis, including 'itself'.

This could make sense only if the ellipsis did not refer to precisely defined terms, but the terms are precisely defined as half the previous term, all the way to infinity.

Whether there's such a thing as "precisely defined infinity" is the whole point of the debate!
You cannot merely assert that the axiom refers to a series that is precisley defined all the way to infinity, because in doing so you are simply begging the question! You are simply asserting that which has yet to be proved.

In fact, this goes to the very heart of my objection! I think it has had more attention than it deserves.

Not at all. I can hardly blame you for making the same mistake as everyone else who thinks that the math are valid, but there is a clear violation of logic inherent within the utilisation of said axiom.

As I said, the only way to equate these two series of numbers, is in finite form. The mere utilisation of an ellipsis does not prove that it is possible to precisely define an infinite series. And anything that cannot be precisley defined, has no equivalence, even with itself.

[The Dagda]

Infinitessimals aren't real any more than infinities hence the limit is what is really happening and is asymptotic or 0 at lim infinity.

When you say that infinitessimals aren't real, are you implying that the world is constituted of discrete elements and that there is a smallest value for each of these elements?

According to empiricism/materialism the smallest probable entity is a quanta.

The smallest possible concept is mathematical and cannot be proven by empiricism.

the mass of photons for example has a lower limit of ~1x10^-60kg

I need to be clear on what you are saying, for reasons that will become apparent. Are you saying that there ARE fundamental particles which are not reducible to anything else? And are you saying that there are fundamental units of space/time/spacetime that are also not reducible to anything smaller? It certainly seems so, but I'd rather have you confirm this, unambiguously, before I proceed.

Yeah energy it is called. E=mc^2.

A particle of light or quanta of energy may be considered as both energy and particle as far as we know all matter can be converted into photons (and may well be at the "end" of the Universe), so they appear to be fundamental. Quarks are partons and do not appear to exist discretely at least at energy levels currently experienced.

[jamest]

Not only that. Someone who can do math will see that the line y - 2x + 3 = 0 is identical to the line 2x - y - 3 = 0.

Isn't that a bit like equating [1/2; 1/4; 1/8; 1/16] with [0.5; 0.25; 0.125; 0.0625]?

I don't see what your point is.

That two such different-looking expressions exactly describing an unending line should be identical in all respects to the unending line with which they are identified is only a puzzle to someone puzzled by the concept of identity, z = z.

For someone to toss out the identity axiom in order to try to establish a new metaphysics is a radical step, and we can see where it has led in this case.

This isn't just about identity. It's about whether anything can have a precise identity as an infinite 'entity'.

[The Dagda]

Not only that. Someone who can do math will see that the line y - 2x + 3 = 0 is identical to the line 2x - y - 3 = 0.

Isn't that a bit like equating [1/2; 1/4; 1/8; 1/16] with [0.5; 0.25; 0.125; 0.0625]?

I don't see what your point is.

That two such different-looking expressions exactly describing an unending line should be identical in all respects to the unending line with which they are identified is only a puzzle to someone puzzled by the concept of identity, z = z.

For someone to toss out the identity axiom in order to try to establish a new metaphysics is a radical step, and we can see where it has led in this case.

This isn't just about identity. It's about whether anything can have a precise identity as an infinite 'entity'.

No only in maths. Few pure mathematicians consider infinity anything more than a concept let alone scientists.

It would help you immensely if you took a crash course in calculus as well.

[GrahamH]

That is: (1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 +....) is only true for definite/finite points of this series. The point being that since 'infinity' is an unknown quantity - and is neither definite nor finite - that the utilisation of the ellipsis (... ) means that there is no equivalence to anything beyond that which is definite or finite. I.e., one cannot equate anything with an ellipsis with anything else with an ellipsis, including 'itself'.

This could make sense only if the ellipsis did not refer to precisely defined terms, but the terms are precisely defined as half the previous term, all the way to infinity.

Whether there's such a thing as "precisely defined infinity" is the whole point of the debate!
You cannot merely assert that the axiom refers to a series that is precisley defined all the way to infinity, because in doing so you are simply begging the question! You are simply asserting that which has yet to be proved.

In fact, this goes to the very heart of my objection!

Not so. The question to be answered is whether a sequence of progressive half steps ever amounts to a finite distance, as defined by Zeno.
We can't calculate a value for a term at infinity, but we know what the term is, and it is identical in all infinite series arising from the definition.

I think it has had more attention than it deserves.

Not at all. I can hardly blame you for making the same mistake as everyone else who thinks that the math are valid, but there is a clear violation of logic inherent within the utilisation of said axiom.

The violation of logic is your implicit claim that A=A is false.

As I said, the only way to equate these two series of numbers, is in finite form. The mere utilisation of an ellipsis does not prove that it is possible to precisely define an infinite series. And anything that cannot be precisley defined, has no equivalence, even with itself.

I can only assume you think "precisely defined" means "calculated", but this is mathematics, not counting.

[jamest]

Infinitessimals aren't real any more than infinities hence the limit is what is really happening and is asymptotic or 0 at lim infinity.

When you say that infinitessimals aren't real, are you implying that the world is constituted of discrete elements and that there is a smallest value for each of these elements?

According to empiricism/materialism the smallest probable entity is a quanta.

The smallest possible concept is mathematical and cannot be proven by empiricism.

the mass of photons for example has a lower limit of ~1x10^-60kg

I need to be clear on what you are saying, for reasons that will become apparent. Are you saying that there ARE fundamental particles which are not reducible to anything else? And are you saying that there are fundamental units of space/time/spacetime that are also not reducible to anything smaller? It certainly seems so, but I'd rather have you confirm this, unambiguously, before I proceed.

Yeah energy it is called. E=mc^2.

Then later, I'll try to explain why this makes no sense.

[jamest]

That is: (1/2 + 1/4 + 1/8 + 1/16 + .... = 1/2 + 1/4 + 1/8 + 1/16 +....) is only true for definite/finite points of this series. The point being that since 'infinity' is an unknown quantity - and is neither definite nor finite - that the utilisation of the ellipsis (... ) means that there is no equivalence to anything beyond that which is definite or finite. I.e., one cannot equate anything with an ellipsis with anything else with an ellipsis, including 'itself'.

This could make sense only if the ellipsis did not refer to precisely defined terms, but the terms are precisely defined as half the previous term, all the way to infinity.

Whether there's such a thing as "precisely defined infinity" is the whole point of the debate!
You cannot merely assert that the axiom refers to a series that is precisley defined all the way to infinity, because in doing so you are simply begging the question! You are simply asserting that which has yet to be proved.

In fact, this goes to the very heart of my objection!

Not so. The question to be answered is whether a sequence of progressive half steps ever amounts to a finite distance, as defined by Zeno.

But the answer that has been provided was done so using an axiom that assumed that an infinite series is "precisely defined all the way to infinity". As I have said, this assumption is debatable and therefore cannot be utilised in the subsequent proof.

The violation of logic is your implicit claim that A=A is false.

Actually, I am saying that it is false where A cannot be precisely defined. Note the distinction.

As I said, the only way to equate these two series of numbers, is in finite form. The mere utilisation of an ellipsis does not prove that it is possible to precisely define an infinite series. And anything that cannot be precisley defined, has no equivalence, even with itself.

I can only assume you think "precisely defined" means "calculated", but this is mathematics, not counting.

No I do not think that. Defining the series all the way to infinity would involve explicitly stating the totality of those terms. I am contending that it is debatable whether such a thing can be done - you cannot simply assert that it can be done as a basis for further mathematical work.

[jamest]

This isn't just about identity. It's about whether anything can have a precise identity as an infinite 'entity'.

No only in maths. Few pure mathematicians consider infinity anything more than a concept let alone scientists.

But one has to consider infinity as more than a concept in order to discuss infinities associated with 'reality'.