Refuting the counters to Zeno's paradox

[jamest]

That's a shame, I think, since I know that you believe in the reality of 'separation'.

What ever gave you that idea?

Well, if you thought that 'reality' was actually One, you'd have to be an idealist.

[The Dagda]

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.
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 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'.

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.

No it doesn't. It is a a philosophical concern only. This is resolved by all mathematical axioms and by reality or science at least.

The interconectedness and distance issues are irellevent in a relativistic time-space system, nothing can travel faster than light, that is all you need to know. Space is not cuttable infinitely any more than time is, except as a concept of infinitessimal concern that does not touch on reality.

Axioms of choice exist because we define x as being equal to x for all cases of x, if we don't then it is not an axiom and is a mere curiosity not worth our attention except to vaguely suppose a universe where 1+1=3. the laws of integration are one such axiom that disproves the validity of ZP, there are plenty of others.

In cantors continuum hypothesis < (integers are "smaller" than the real numbers because they are not 1 to 1) that doesn't mean that if we apply that to a real co-ordinate system the space between them is bigger only the conceptual co-ordinate system contains more terms. the same can be said of Zenos paradox which is limited by the non infinite nature of both space and time. No matter how finely you cut up space, there is no more distance between an object at time t than there can finitely exist. That's it /thread.

[GrahamH]

... 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:

Why would the equivalences between identical infinite series be finite? The opposite of this is used in the proof that the set of even numbers is infinite, by pairing each even number with an integer.

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.
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 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'.

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.

You are challenging the validity of mathematics here. 1/2 + 1/4 + 1/8 + 1/16 +.... is defined as an infinite series of definite form. You would need to show that there is some invalidity to using labels (such as S or 1/2 + 1/4 + 1/8 + 1/16 +....) to represent infinite series.

Your objections don't look good to me. I admit that I'm not a mathematician, but neither are you, and your attack here is on the foundations of mathematics.

[The Dagda]

... 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:

Why would the equivalences between identical infinite series be finite? The opposite of this is used in the proof that the set of even numbers is infinite, by pairing each even number with an integer.

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.
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 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'.

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.

You are challenging the validity of mathematics here. 1/2 + 1/4 + 1/8 + 1/16 +.... is defined as an infinite series of definite form. You would need to show that there is some invalidity to using labels (such as S or 1/2 + 1/4 + 1/8 + 1/16 +....) to represent infinite series.

Your objections don't look good to me. I admit that I'm not a mathematician, but neither are you, and your attack here is on the foundations of mathematics.

I am not a mathemetician more of a person studying physics and yes he is wrong as Cantors continuum adequately demonstrates its possible to conceive of a number of infinities in a continuum but it has only philosophical merit.

the set of Z and R and aleph alpha to omega are conceptual concerns not space-time ones.

[The Dagda]

Assuming the axiom of choice, there is a smallest cardinal number \aleph_1 greater than \aleph_0, and the continuum hypothesis is in turn equivalent to the equality



There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis (GCH) which says that for all ordinals \alpha\,

[The Dagda]

A consequence of the hypothesis is that every infinite subset of the real numbers either has the same cardinality as the integers or the same cardinality as the entire set of the reals.

Stupid damn posting protocols!

I wish VBB would encode latex as a part of V2.0 onwards.

http://en.wikipedia.org/wiki/Continuum_hypothesis

[Surendra Darathy]

I am not a mathemetician more of a person studying physics and yes he is wrong as Cantors continuum adequately demonstrates its possible to conceive of a number of infinities in a continuum but it has only philosophical merit.

To be fair to James, he is talking about "Reality" rather than about mathematics or physics. James is trying to make a philosophical point, that if "Reality is One", then physics is strictly an invented system to deal with empirical experience and mathematics is only another human language.

Continuing to be fair, some mathematicians and physicists will explicitly state their own philosophical position with respect to mathematics and/or physics. Penrose, for example, does this in the first chapter of The Road to Reality. On p, 19, Penrose writes:

...On this view, everything in the universe is indeed governed in completely precise detail by mathematical principles — perhaps by equations, such as those we shall be learning about in chapters to follow, or perhaps by some future mathematical notions fundamentally different from those which we would today label by the term 'equations'. If this is right, then even our own physical actions would be entirely subject to such ultimate mathematical control, where 'control' might still allow for some random behavior governed by strict probabilistic principles.

Many people feel uncomfortable with contentions of this kind, and I must confess to having some unease with it myself. Nonetheless, my personal prejudices are indeed to favour a viewpoint of this general nature, since it is hard to see how any line can be drawn to separate physical actions under mathematical control from those which might lie beyond it. In my own view, the unease that many readers may share with me on this issue partly arises from a very limited notion of what 'mathematical control' might entail.

Note that Penrose carefully and explicitly denotes personal opinion wherever he refers to it. Penrose also expresses his philosophical position about the ontology of mathematical objects, viz., that they exist independently of human formulations of them. But he carefully and explicitly describes this as his personal position and notes that other positions in regard to this are possible. That is the difference between doing philosophy and doing mathematics.

James insists (without much bother that it is merely a personal conviction of his) that arguments citing mathematics and physics are irrelevant in sorting out the philosophical nature of Zeno's Paradox:

Fundamentally, Zeno questions the reality of objects that are somehow distanced from one another. Now, essentially, it doesn't really matter which parameters we use to explain this separation - the bottom-line is that whichever parameters correctly describe the separation between objects must be quantifiable. Therefore, those parameters must be subject to the kind of reasoning that Zeno utilised.

Sure, Zeno's understanding of space & time might appear relatively simple, compared to how we describe spacetime, now. But what we have to consider, is that Zeno is essentially questioning the reality of separation. It doesn't matter, then, how he or we [now] describe this separation - that's missing the point entirely. Ultimately, the fundamental point of Zeno's reasoning has to be taken seriously and we have to explain how quantifiable measures of separation can exist between objects in reality.

Of course, since it is questionable that what is being observed cannot be 'real', then anything that physics says cannot really be employed to counter Zeno's argument, as observation is the basis for whatever physics might say. So it seems to me as though this discussion has taken a turn for the worse, with several people trying to refute Zeno via current physics.

Physics and mathematics are constrained by (in the first case) empirical experience and (in the second case) by sets of axioms clearly defined and expressed, whereas metaphysics simply involves making up whatever shit sounds good to you.

If Zeno has presented reasoning that questions the reality of motion/space, then what's your problem with my statement?

If a metaphysical question on Reality comes up, it is automatically valid.

Nobody (even Zeno) can question the reality of space or motion without questioning the reality of objects that are somehow distanced from one another. Therefore, I don't need references to say this.

James is asserting that making shit up is a valid mode of discourse. Otherwise, religion itself would be impossible.

You are challenging the validity of mathematics here. 1/2 + 1/4 + 1/8 + 1/16 +.... is defined as an infinite series of definite form. You would need to show that there is some invalidity to using labels (such as S or 1/2 + 1/4 + 1/8 + 1/16 +....) to represent infinite series.

Your objections don't look good to me. I admit that I'm not a mathematician, but neither are you, and your attack here is on the foundations of mathematics.

James is mounting his attack on mathematics, physics, and rational discourse, consisting of rectally-extracted assertions that the castle walls are crumbling. The fact that his objections don't look good to you is entirely irrelevant. Reality is One! and James is a prophet dishonored in his own country.

Anyone (even Penrose!) may entertain and express philosophical convictions, including an entirely incoherent set, as James has expressed for us. This is human perversity in action, and a possible explanation for the existence of billions of woo-heads. When a confusion exists between personal conviction and discursive argument, the result is the sort of performance that James has given us.

I think that James is entitled to announce his personal opinions, and even to be sure of himself that these opinions must also convince you and me. It's silly, I know, but that is one of the wonders of the internet.

[The Dagda]

oh yeah I know he's talking philosophically but such concerns do not matter and are in themselves just saying the axioms are wrong and lead nowhere without proving why they are wrong. How do you do that?

[jamest]

oh yeah I know he's talking philosophically but such concerns do not matter and are in themselves just saying the axioms are wrong and lead nowhere without proving why they are wrong. How do you do that?

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.

Regarding another thing that you have said in a previous post:

The interconectedness and distance issues are irellevent in a relativistic time-space system, nothing can travel faster than light, that is all you need to know. Space is not cuttable infinitely any more than time is, except as a concept of infinitessimal concern that does not touch on reality.

I believe that something else I said on page 13 answers this concern:

Fundamentally, Zeno questions the reality of objects that are somehow distanced from one another. Now, essentially, it doesn't really matter which parameters we use to explain this separation - the bottom-line is that whichever parameters correctly describe the separation between objects must be quantifiable. Therefore, those parameters must be subject to the kind of reasoning that Zeno utilised.

Sure, Zeno's understanding of space & time might appear relatively simple, compared to how we describe spacetime, now. But what we have to consider, is that Zeno is essentially questioning the reality of separation. It doesn't matter, then, how he or we [now] describe this separation - that's missing the point entirely. Ultimately, the fundamental point of Zeno's reasoning has to be taken seriously and we have to explain how quantifiable measures of separation can exist between objects in reality.

Of course, since it is questionable that what is being observed cannot be 'real', then anything that physics says cannot really be employed to counter Zeno's argument, as observation is the basis for whatever physics might say. So it seems to me as though this discussion has taken a turn for the worse, with several people trying to refute Zeno via current physics.

You know, I can make an argument that the relativistic nature of space-time excludes its reality beyond that of the mind that perceives their values. And since any supposed counter to Zeno must intend to prove the 'reality' of the aforementioned dimensions, then what you say could be categorised as self-defeating, anyway. Regardless, as reasoned, above, nothing physics says can be employed to counter Zeno.

[The Dagda]

oh yeah I know he's talking philosophically but such concerns do not matter and are in themselves just saying the axioms are wrong and lead nowhere without proving why they are wrong. How do you do that?

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.

Regarding another thing that you have said in a previous post:

The interconectedness and distance issues are irellevent in a relativistic time-space system, nothing can travel faster than light, that is all you need to know. Space is not cuttable infinitely any more than time is, except as a concept of infinitessimal concern that does not touch on reality.

I believe that something else I said on page 13 answers this concern:

Fundamentally, Zeno questions the reality of objects that are somehow distanced from one another. Now, essentially, it doesn't really matter which parameters we use to explain this separation - the bottom-line is that whichever parameters correctly describe the separation between objects must be quantifiable. Therefore, those parameters must be subject to the kind of reasoning that Zeno utilised.

Sure, Zeno's understanding of space & time might appear relatively simple, compared to how we describe spacetime, now. But what we have to consider, is that Zeno is essentially questioning the reality of separation. It doesn't matter, then, how he or we [now] describe this separation - that's missing the point entirely. Ultimately, the fundamental point of Zeno's reasoning has to be taken seriously and we have to explain how quantifiable measures of separation can exist between objects in reality.

Of course, since it is questionable that what is being observed cannot be 'real', then anything that physics says cannot really be employed to counter Zeno's argument, as observation is the basis for whatever physics might say. So it seems to me as though this discussion has taken a turn for the worse, with several people trying to refute Zeno via current physics.

You know, I can make an argument that the relativistic nature of space-time excludes its reality beyond that of the mind that perceives their values. And since any supposed counter to Zeno must intend to prove the 'reality' of the aforementioned dimensions, then what you say could be categorised as self-defeating, anyway. Regardless, as reasoned, above, nothing physics says can be employed to counter Zeno.

Well I'm pretty sure you don't do calculus so wasting my time on proofs that involve it are superfluous. If you can't understand why the base rules of calculus make Zenos paradox an absurdity then I can't help you, Aristotle is responsible for the proof and he's dead.

Your maths kinda sucks on this issue at least, what level are you at?

I could wax lyrical about space time and special relativity and the constraints of matter by the speed limit of the universe and s=d/t hence therein but that too would be a waste of time as you're trying to attack axioms that have no application to the real world and are abstractions. 1 is not a physical thing it is a concept.

To prove yourself correct you'd need to show either in Sum or integral form why the axioms are wrong, I know what your concern is I just don't think it really is one. it's a non issue.

1) If x=y
2) then z is false

Very true but it doesn't x=x.

We do explain distance it is defined as a number of hyperfine oscillations of a caesium atom and how fast light travels in a vacuum or c, it is known as m/s it is constrained by special relativity and the axiom is that all experiment agrees that space time is limited by the Lorentzian transforms assuming c is a constant.

As I said dividing 1/x has as solution only if we assume in all cases of x the limit is infinity and = 0.

subbing into the equations of physics leaves ultimately the same result as the equation and is merely a superfluous irrelevancy of relativity.



Essentially you are saying what if x=y and in that case you are wrong x=x.

[jamest]

Well I'm pretty sure you don't do calculus so wasting my time on proofs that involve it are superfluous.

As I said, I've just focussed upon the initial axiom of one particular proof. As you have seen, I have attempted to undermine the basis of that math, rather than prove that the math in itself is wrong, per se. You're right, I don't do calculus. So presenting a full proof using calculus would probably go over my head. It might be interesting to discuss the initial premises of that proof, though. So if you want to explain what the proof is grounded upon, feel free.

I could wax lyrical about space time and special relativity and the constraints of matter by the speed limit of the universe and s=d/t hence therein but that too would be a waste of time as you're trying to attack axioms that have no application to the real world and are abstractions. 1 is not a physical thing it is a concept.

Again, equating the world with 'reality' is the questionable aspect of this debate. I've tried explaining this to you - and why anything physics says is not relevant.

[The Dagda]

Well I'm pretty sure you don't do calculus so wasting my time on proofs that involve it are superfluous.

As I said, I've just focussed upon the initial axiom of one particular proof. As you have seen, I have attempted to undermine the basis of that math, rather than prove that the math in itself is wrong, per se. You're right, I don't do calculus. So presenting a full proof using calculus would probably go over my head. It might be interesting to discuss the initial premises of that proof, though. So if you want to explain what the proof is grounded upon, feel free.

I could wax lyrical about space time and special relativity and the constraints of matter by the speed limit of the universe and s=d/t hence therein but that too would be a waste of time as you're trying to attack axioms that have no application to the real world and are abstractions. 1 is not a physical thing it is a concept.

Again, equating the world with 'reality' is the questionable aspect of this debate. I've tried explaining this to you - and why anything physics says is not relevant.

The proof is grounded in summation and integral maths though. A series of potentially infinite terms is a Taylor series that converges as x approaches infinity.

f(x)= ln(x)
f'(x)=1/x

Definite solutions arrive at exact measurements.

Basically the change on a graph of ln(x) or rise over run (speed/time if you want) =1/x.

the above are axioms of choice and hence proofs because they are self consistent, they also agree with science which is nice.

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


There are not infinite sub divisions as time is relative but distance need not be.

Have you hear of the time dependant Schrödinger equation or the general time independent equations.

[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?

[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

[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. 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.