Refuting the counters to Zeno's paradox

[GrahamH]

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.

There is no need to specify the totality of those terms unless you want to add them up. They can be defined with one simple expression.

The confusion comes from arriving at a term one over infinity. Any finite value divided by infinity is not distinguishable from any other, so identical equalities don't apply. But this ignores the fact that we know the definition of the terms in this case is identical. It is half the previous term, by definition. That definition doesn't change, all the way to infinity. We can't calculate all the values, but we do know how they are defined, so the terms are equal, whatever their numerical values might actually be. To counter that you would have to show that the definition of the terms changes, but it doesn't.

[jamest]

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.

There is no need to specify the totality of those terms unless you want to add them up.

If you want to equate A with itself, then you have to show that A can be precisely defined. If A cannot be precisely defined, then it makes no sense to say that A = A. It is meaningless.

[GrahamH]

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.

There is no need to specify the totality of those terms unless you want to add them up.

If you want to equate A with itself, then you have to show that A can be precisely defined. If A cannot be precisely defined, then it makes no sense to say that A = A. It is meaningless.

What is undefined in the definition of the terms of the series? The definition of all the terms is half the value of the previous term. No term deviates from that. How is it not a precise definition?

[jamest]

What is undefined in the definition of the terms of the series? The definition of all the terms is half the value of the previous term. No term deviates from that. How is it not a precise definition?

That is a definition of the relationship inherent between all of the terms [of the series] - that definition doesn't suffice to completely (key word) define 'the entity' itself. Indeed, that same definition would apply to a finite series of the same numbers! Your definition clearly requires amendment to distinguish the infinite series from a finite series.

[Surendra Darathy]

If you want to equate A with itself, then you have to show that A can be precisely defined. If A cannot be precisely defined, then it makes no sense to say that A = A. It is meaningless.

Your personal incapacity to deal with mathematical (and precise) definitions of infinity and infinitesimals by trying to "conceptualise" them does not render it meaningless, simply because a lot of maths can be expressed by using infinity and infinitesimals according to their definitions.

There is no lower bound for a sequence of positive real numbers generated by halving the previous element in the sequence. This does not mean that the next element in the sequence is undefined.

The mathematical definition of an infinite series permits one to calculate the value of the summation when the series converges. The series represented by the trajectory of Zeno's arrow converges to a defined value. Your incapacity to apply definitions of infinity and infinitesimals does not invalidate a big chunk of mathematical analysis.

Instead, it makes you into someone shouting, "Everybody's out of step but me," when you have no parade in sight. You're simply mistaking the question of whether mathematical objects "exist" with the question of whether or not mathematics is applicable to physics. Since mathematical physics permits the prediction of the outcome of experiments, we say that mathematics is applicable to physics.

[The Dagda]

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

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

Oh dear I'm not looking forward to that. Quantum mechanics destroyed by someone who has no knowledge of physics or maths. I'd say there would be a certain amount of schardenfreuder in listening to ill informed musings on interpretation (philosophical issues) but there really isn't this is not a subject for novice mathematicians or philosophers, except of science.

Be sure to start with the difference between energy and matter, it's not as obvious as it sounds.

[GrahamH]

What is undefined in the definition of the terms of the series? The definition of all the terms is half the value of the previous term. No term deviates from that. How is it not a precise definition?

That is a definition of the relationship inherent between all of the terms [of the series] - that definition doesn't suffice to completely (key word) define 'the entity' itself. Indeed, that same definition would apply to a finite series of the same numbers! Your definition clearly requires amendment to distinguish the infinite series from a finite series.

Not so, the definition and terms are identical both cases, but the limits are different

Can't type proper notation here, but this should be clear enough...

X = SUM (1/2n), n=1 -> infnity

X = SUM (1/2n), n=1 -> 199

The same precise definition applies to all terms in the series.

This is pairing of terms to prove equality of an infinite series is very like the proof that the even and odd numbers are an infinite set. See here. There is no term in the series that does not pair, and each has the same definition to, and so must be equal to, its pair. Therefore the series in the proof are identical and the substitution is valid.

You probably should learn some calculus. It isn't required for the solution to Zeno, but might be helpful if you want to think about infinite series and infinitesimals in more general terms. I guess you can then come back and tell us all how Newton and Leibniz got it all wrong.

[Xamonas Chegwé]

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.

There is no need to specify the totality of those terms unless you want to add them up.

If you want to equate A with itself, then you have to show that A can be precisely defined. If A cannot be precisely defined, then it makes no sense to say that A = A. It is meaningless.

This is a very naïve statement. Equality is axiomatic. By its very definition any expression is equal to itself - that is, equality possesses the reflexive property.

The phone number of the next person to call you = The phone number of the next person to call you

is a valid equality even if you miss the call and don't know it was made, or if even you have no telephone!

Just because no precise definition can be given to a value DOES NOT invalidate its self-equality.

[Surendra Darathy]

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

What you should try for is to explain why it makes no sense to you. A trouble-free "explanation" will involve your admission of not understanding the first thing about maths and physics.

But you could add to this your definition of what constitutes an "explanation". If you're making an answer that you decide a priori should be satisfactory to everyone else, you're making what is otherwise known as an "excuse". In this case, it is your "excuse" as to the lack of relevance of physics and mathematics to anyone's thinking, let alone merely your own.

One of the experiences that people seem to like is the situation of agreement. Empirical science is better at achieving this state than medieval wibbling about Platonic forms, or what-have-you. That's what we mean by "better".

Explaining "why this makes no sense" is a rationale for your state of disagreement, not a means of being able to do anything about it.

[GrahamH]

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.

There is no need to specify the totality of those terms unless you want to add them up.

If you want to equate A with itself, then you have to show that A can be precisely defined. If A cannot be precisely defined, then it makes no sense to say that A = A. It is meaningless.

This is a very naïve statement. Equality is axiomatic. By its very definition any expression is equal to itself - that is, equality possesses the reflexive property.

The phone number of the next person to call you = The phone number of the next person to call you

is a valid equality even if you miss the call and don't know it was made, or if even you have no telephone!

Just because no precise definition can be given to a value DOES NOT invalidate its self-equality.

I wonder if james is confusing the A = A thing with the problem of equating different things involving infinities.

n/∞ = m/∞ does not mean that n = m

n/∞ = n/∞, although it is not calculable it is the same by definition.

What do you think?

[Surendra Darathy]

I wonder if james is confusing the A = A thing with the problem of equating different things involving infinities.

n/∞ = m/∞ does not mean that n = m

n/∞ = n/∞, although it is not calculable it is the same by definition.

If we are computing a limiting value for a quotient of the form m/p, where m and p are real expressions, and m/p reduces to

∞/∞. 0/∞, or ∞/0,

additional steps may be required to determine whether the quotient has a definite value or not in terms of real analysis.

I don't know much about algebraic versions for fields such as rationals or reals where division is defined, but division by zero is not.

So, there may be a confusion between treating Zeno algebraically as opposed to analytically. Of course, James will exclude a mathematical treatment as relevant, just as he does a physics (kinematics) analysis.

As originally posed, Zeno's paradox is nonsense if the arrow is assumed to have spatial extent, but the trajectory is not. We have then to go all the way back to the beginning and propose that we are talking about the illusion that illusions can engage in motion, a question so absurd that not even a metaphysician will touch it.

[The Dagda]

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

What you should try for is to explain why it makes no sense to you. A trouble-free "explanation" will involve your admission of not understanding the first thing about maths and physics.

But you could add to this your definition of what constitutes an "explanation". If you're making an answer that you decide a priori should be satisfactory to everyone else, you're making what is otherwise known as an "excuse". In this case, it is your "excuse" as to the lack of relevance of physics and mathematics to anyone's thinking, let alone merely your own.

One of the experiences that people seem to like is the situation of agreement. Empirical science is better at achieving this state than medieval wibbling about Platonic forms, or what-have-you. That's what we mean by "better".

Explaining "why this makes no sense" is a rationale for your state of disagreement, not a means of being able to do anything about it.

QFT.

Seriously I respect any question honestly asked, and any admission of lack of knowledge. But you do need that knowledge for this to be anything more than a moot argument. People like the above like the cut and thrust of mathematical debate but it's difficult for them - not me I'm but a student of maths - to educate the laity in simple mathematical concepts, when they clearly don't have the education to understand any of the points made. That's not your fault, but it is your weakness.

[Comte de Saint-Germain]

Seriously I respect any question honestly asked, and any admission of lack of knowledge.

Precisely!

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.

There is no need to specify the totality of those terms unless you want to add them up.

If you want to equate A with itself, then you have to show that A can be precisely defined. If A cannot be precisely defined, then it makes no sense to say that A = A. It is meaningless.

This is a very naïve statement. Equality is axiomatic. By its very definition any expression is equal to itself - that is, equality possesses the reflexive property.

The phone number of the next person to call you = The phone number of the next person to call you

is a valid equality even if you miss the call and don't know it was made, or if even you have no telephone!

Just because no precise definition can be given to a value DOES NOT invalidate its self-equality.

Excellent explanation! Thanks.

[jamest]

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.

There is no need to specify the totality of those terms unless you want to add them up.

If you want to equate A with itself, then you have to show that A can be precisely defined. If A cannot be precisely defined, then it makes no sense to say that A = A. It is meaningless.

This is a very naïve statement. Equality is axiomatic. By its very definition any expression is equal to itself - that is, equality possesses the reflexive property.

This is the point of contention. Asserting your belief isn't very helpful, especially when you haven't addressed any of the reasons I have presented to counter that view. Oh not it isn't... oh yes it is - such debates are pointless. I'm trying very-hard to provide reasons to challenge the assumption that you have utilised, so I would appreciate more than a roll of the eyes as a response.

Ultimately, we must all acknowledge that there is a difference between a finite 'entity' and an infinite entity. Therefore, we must think twice before assuming that the obvious axioms which we apply to finite entities, necessarily apply to infinite entities - especially in a discussion such as this, which seeks to justify the reality of infinite 'entities'. It seems that I have to reiterate, yet again, that this is supposed to be a rational discussion about your justification for the utilisation of that initial axiom, as the basis for your subsequent math... and that this is now, in no sense, a mathematical discussion. You simply cannot prove/justify, mathematically, that A = A for all entities, including infinite ones. If you could, A = A wouldn't be the BASIS for your argument.

You're just asserting that which you need to be true, in order to make your subsequent conclusion. The very point that this a rational issue should alert you to the dubiousness of your claim, which means that you should be trying to justify it, rather than just parroting it.

I put it to you that there can be no equivalence of anything that itself has no definite form. What is the equivalence of incompleteness? 'A' can only be mirrored in its [finite] completeness... otherwise, it has no mirror-image.

What is '1'? It's a complete and definite concept. So how can something that is forever incomplete ever be completed? How can an UNENDING and therefore incomplete entity, ever be completed? It simply does not compute to say that it can... and your mathematical conclusion DOES complete an unending series. That's why it makes no sense... and that's why the conclusion is at-odds with this understanding. Indeed, this understanding alone such alert you to the fact that a mistake must have been made, in coming to that conclusion.

Please don't conflate an infinite entity with an eternally-expanding finite-entity, since an eternally-expanding finite entity is always a finite entity, no matter the potential for its extensiveness. The point of this discussion, is to seek an understanding of the reality of infinite 'entities' as distinct to finite entities. It is not to seek an understanding of the the distinction between a finite entity and a finite entity that never stops expanding in size.

I put it to you that A = A, is correct just for finite entities - even for those that forever expand in definite/finite form. But if 'A' is defined, ultimately, as that which is not finite, then we have a problem. That is, infinite entities are never completely defined, and, therefore, cannot expand. For, how can something that is without complete/finite definition, grow beyond what it is?

I contend that the utilisation of your initial axiom was a confused consequence of conflating expanding finiteness with 'infiniteness' - of applying the same intuitive knowledge we have of finiteness, with infiniteness. And I contend that you have no reason to justify this conflation of concepts.

The phone number of the next person to call you = The phone number of the next person to call you

This is irrelevant and irrational, as the number of the next person to call me will have to be finite, if I ever hope to call him back. Even in eternity. Indeed, if the phone number of the person that calls me is infinite, then it is illogical that I should ever receive a phone call... since that implies that he pressed a final digit!

is a valid equality even if you miss the call and don't know it was made, or if even you have no telephone!

If I have no telephone, I receive no telephone call.

[jamest]

Seriously I respect any question honestly asked, and any admission of lack of knowledge.

Precisely!

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.

There is no need to specify the totality of those terms unless you want to add them up.

If you want to equate A with itself, then you have to show that A can be precisely defined. If A cannot be precisely defined, then it makes no sense to say that A = A. It is meaningless.

This is a very naïve statement. Equality is axiomatic. By its very definition any expression is equal to itself - that is, equality possesses the reflexive property.

The phone number of the next person to call you = The phone number of the next person to call you

is a valid equality even if you miss the call and don't know it was made, or if even you have no telephone!

Just because no precise definition can be given to a value DOES NOT invalidate its self-equality.

Excellent explanation! Thanks.

Clearly, you missed the logical errors too.