Quite a few years ago I took a accumulation of A akin aesthetics acceptance to a baby university in Wales which was agog to advance its aesthetics department.
Presentations on a amount of abstract capacity were accustomed to A akin acceptance from assorted schools and sixth anatomy colleges possibly absorbed in advancing a amount in philosophy. A part of these was an annual of the abstract botheration of what is alleged Zeno's paradox, which 'proves' that movement is impossible. Aback acutely movement is possible, the 'proof' is paradoxical.
You are allegedly able-bodied acquainted of the story, but I'll present it briefly in case it would be helpful. A tortoise and Achilles accept a race. To accomplish it added fair, the tortoise is accustomed a arch start. But, actually, the tortoise is not accustomed abundant of a advance and, in practice, it will be simple for Achilles to bolt up and beat it. But he can't anytime do this, according to the paradox. The botheration is that, in adjustment to beat the tortoise, Achilles will aboriginal accept to ability the point area the tortoise started. But if Achilles alcove this atom the tortoise will accept gone added forward. Achilles afresh has to get to the tortoise's new position. But if he gets there the tortoise will afresh accept confused forward. This continues indefinitely - or, rather, infinitely. Anniversary time Achilles alcove area the tortoise had got to, the tortoise has confused on. Achilles can accordingly - the altercation goes - never bolt up with the tortoise.
Another way of putting the botheration is that, to ability the tortoise, Achilles has to get half-way to area the tortoise is at any accustomed point. And already Achilles alcove the half-way point, he afresh has to ability half-way to area the tortoise now is. And if he alcove this point, he afresh has to... I anticipate you get the point. This happens to infinity, so Achilles gets afterpiece and afterpiece to but never catches up with the tortoise.
In fact, according to this logic, no movement is accessible at all because, in adjustment to move to the tortoise's starting point, Achilles has to get half-way there. And in adjustment to get half-way there, he has to get half-way to this point. But in adjustment to get to this point he has to get half-way there too. And so on. In added words, Achilles has to biking through an beyond of credibility even to get started. Aback he can't do this, he can't move at all - and nor can anybody or annihilation else.
I was placed a part of a accumulation of acceptance from assorted schools accustomed the assignment of discussing this problem. It happened that the academician who presented Zeno's absurdity was in my group. I appropriate that, if beyond doesn't exist, the absurdity doesn't abide and Zeno's botheration vanishes as a mirage. My advancement was swept abreast with the words, 'You can't just abolish infinity!'
Logically, with account to the absurdity there are three possibilities: (1) that movement is not possible; (2) that beyond does not exist; (3) that Zeno's absurdity is adulterated philosophy. The aboriginal achievability can confidently be rejected. The third can aswell provisionally be rejected, aback abounding philosophers still attention Zeno's altercation as pertinent. This leaves us with the additional possibility: that beyond does not exist. This achievability has not to my ability been anxiously considered.
In this article, I'd like to appearance that beyond doesn't exist. Without infinity, there is no absurdity for Zeno: if there are not an absolute amount of credibility to traverse, but just a specific distance, afresh there is no botheration in authoritative the aboriginal step, in accepting half-way to anywhere, or in communicable up with and overtaking a slower affective article (given acceptable time).
I accept a amount of arguments to prove that beyond doesn't abide - which are conceivably all the aforementioned altercation airish in altered ways.
1 Consistently small
An consistently baby assemblage is an impossibility. For a assemblage to be consistently baby it would agree with zero. How big is an consistently baby space? It's no amplitude at all, because if it took up any amplitude whatsoever it would not be consistently small. An consistently baby amplitude accordingly does not exist. Getting able to allocution about an consistently baby unit, as if it has existence, does not accord it achievement - any added than talking about Martians or superheroes, or alongside universes, accomplish them a reality.
Here's addition way of demonstrating the aforementioned point. You ability accept at aboriginal glance that 9.99 alternating is an consistently baby ambit abroad from 10 and represents an consistently baby fraction, but 9.99 alternating is in achievement the aforementioned amount as 10. This is simple to demonstrate. What is 10 disconnected by 3? The acknowledgment is 3.33 recurring. And what is 3.33 alternating assorted by 3? It's 9.99 alternating - or 10. The two numbers are the same.
Since Zeno's absurdity is abased on the angle of consistently baby spaces, this altercation should be abundant to abrogate it. But I'd like to yield the befalling to appearance that beyond has no achievement with account to amplitude as able-bodied as smallness.
2 Consistently large
There is no such affair as an consistently baby unit. But could an consistently ample amplitude exist? Yes, if the cosmos is infinite, it ability be argued. If it's not, an consistently ample amplitude does not and cannot exist. So could the cosmos be infinite? I shall altercate that it can't be.
For the cosmos to be consistently ample there would accept to be two points, let's alarm them brilliant A and brilliant B, that are an absolute ambit apart. You can't accept an absolute cosmos unless there are credibility that are consistently far apart, because if the extreme two credibility are alone a apprenticed ambit afar afresh the cosmos is not infinite.
But as anon as two credibility are anchored in amplitude they accept a spatial accord with anniversary other, and the amplitude amid them cannot be infinite.
Two credibility anchored in amplitude accept to accept a point which is about half-way amid them, which we'll alarm brilliant C. If that abode is assessable as half-way amid them, about astronomic the measurement, the ambit is finite.
You ability wish to altercate that there is an absolute ambit amid brilliant A and this half-way point, brilliant C. Afresh acquisition the point, brilliant D, half-way amid A and C. Is this aswell an absolute ambit from brilliant A? Afresh do the aforementioned again. And again. Accumulate traveling an absolute amount of times (supposing such a affair exists). Are the two credibility still an absolute ambit away? They accept to still and consistently be an absolute ambit away. This agency of advance that there cannot be a half-way point amid stars A and B in an absolute universe.
From the area of brilliant A, in which administration is brilliant B? Brilliant B is in no administration in affiliation to brilliant A if it is an absolute ambit away. If you could bend any afterpiece to it at all, you wouldn't be an absolute ambit away. To be an absolute ambit afar the two credibility can accept no spatial accord with anniversary other. Remember, it's not that any administration will be in the administration of brilliant B (which could be accurate if amplitude were apprenticed and arced in four ambit as the apple is in three dimensions), but that no administration leads in the administration of brilliant B. If they accept no accord with anniversary added - if in affiliation to brilliant A brilliant B is neither up, down, left, right, advanced or astern (or any added believable direction), it has no achievement about to brilliant A.
Travelling in all admonition from brilliant A, you'll never ability anywhere that has a spatial accord with (i.e. is allegedly assessable in affiliation to) brilliant B. Similarly, travelling in all admonition from brilliant B, you'll never ability anywhere that has a spatial accord with brilliant A. Imagine ablaze travelling in all admonition from stars A and B. About far the ablaze travels, it can never ability infinity. A apprenticed ambit can never become an absolute ambit (see altercation # 3 below). The abstraction of an absolute cosmos afield assumes that finity can become beyond aback stars (or credibility in space) are accounted to abide which are in the 'same' cosmos an absolute ambit away.
Star A * [finite distance]... [becomes infinite]... [finite distance] Brilliant B *
There is no acumen to accept (in this abstract absolute universe) that there are alone two 'areas' (star A and brilliant B and all credibility addition indefinitely from both in all directions) which are an absolute ambit apart. How could the amount of such 'areas' in an absolute cosmos be apprenticed to any apprenticed number? There accept to be an absolute amount of locations an absolute ambit apart, all getting like a abstracted cosmos in themselves, accepting no spatial affiliation with the added beyond of 'universes' which are an absolute ambit away.
Star A * [finite distance]... [becomes infinite]
Star B * [finite distance]... [becomes infinite]
But area are the boundaries of these allegedly abstracted (because spatially unconnected) universes? There are no boundaries, aback they go on forever. If they go on consistently and accept no boundaries they accept to be infinite. This agency that there accept to be credibility an absolute ambit away. But we've been through all this before...
Star A * [finite distance]... [becomes infinite]... Brilliant AZ * [an absolute ambit from brilliant A]
Star B * [finite distance]... [becomes infinite]... Brilliant BZ * [an absolute ambit from brilliant B]
As anon as beyond is postulated, beyond proliferates into an beyond of infinities.
But none of this is possible. A apprenticed ambit can never become an absolute distance. Biking from one point as far as you like, you never get an absolute ambit away. There cannot accordingly be two credibility an absolute ambit from anniversary other.
Two credibility advised in amplitude may be a absolute continued ambit apart, but they can't be an absolute distance. As anon as credibility are advised in space, i.e. accustomed a location, the ambit amid them is allegedly measurable. There is no such affair as an consistently ample amplitude aback finity can never become infinity. Amplitude cannot be infinite. Consistently ample is as absurd as consistently small.
If consistently baby units are accounted to exist, there is no achievability of movement. If consistently ample units are accounted to exist, locations in amplitude accept to abide (in absolute number) that accept no accord with added altar afar consistently from them.
3 Traveling on forever
But, you ability object, we can accumulate halving - or acceleration - a amount forever. There is no absolute to how baby or big we can go. This is true, and it gives the apparition that beyond is possible. But although you can accumulate halving or acceleration forever, consistently never arrives: you never get to infinity. You never ability a amount that is an absolute ambit from the amount you started with. About big the amount is, it is still a amount and has a accord with the numbers that appear afore and afterwards it.
This is just like the apprenticed cosmos argument. About far abroad the credibility are, they never ability infinite. Accumulate counting and never stop; calculation in huge leaps; quadruple the amount a abundance times with every count; calculation until the numbers are as continued as a galaxy, as continued as amplitude itself - you'll never get to infinity. Finity never becomes infinity. Very, absolute big numbers are possible, as are very, absolute baby fractions - but not consistently big numbers nor consistently baby fractions.
Infinity is a concept. We accept no agitation alive what it agency (even admitting we generally use it about in an erroneous way, adage for archetype that something is 'almost infinite'). Potentials can be infinite: possibilities are infinite; but a achievability is not an actuality. Beyond has no absolute existence. Annihilation in the cosmos can be infinite. No infinity, no Zeno's paradox. Sorry, Zeno, and aesthetics academician in Wales.
A absurdity charcoal however, apropos not amplitude but time. Time, from the present, can go on consistently and beyond can never be accomplished - an absolute time will never accept passed. That is accurate of the future. But what of the past? If time had no alpha but consistently existed, this would beggarly an beyond has already anesthetized - a angle that is bottomless with attention to space. But if time does not go backwards for ever, it accept to accept started at some point. If the cosmos started at some point it would beggarly that something was created out of nothing: that there was no space, time, or matter, and al of a sudden these things came into being. Either time goes aback for ever, or started at a assertive point. Either is appropriately impossible.
Even if an absolute God exists (not apprenticed by time and space, which are finite) who, at a assertive moment, created space, time and amount from nothing, what was God accomplishing afore this? God, getting infinite, accept to accept been about continued afore the conception of the universe, in the area of beyond - just blind around.
Zeno's absurdity is commonly anticipation to advance the impossible: that, accustomed the achievement of infinity, movement is impossible. But the absolute accent of the absurdity is the beyond proof; or, if you prefer, the finity proof: for any movement in amplitude to be possible, for any action at all to be accessible (since all action involves movement), and for all altar to abide in accord with anniversary added - beyond can accept no existence.