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Deconstructing Infinity: An Analysis of Zeno's Paradox

By gdanjo in Science
Fri Jan 07, 2005 at 08:21:40 AM EST
Tags: Culture (all tags)
Culture

Zeno of Elea famously postulated his many paradoxes in defence of Parmenides' worldview: that all is "oneness," and pluralism is merely an illusion.

Of his forty paradoxes, the four most enduring have fascinated philosophers, mathematicians, and regular pundits for millennia.

As one of the many pundits ensnared by Zeno's challenges, I will present my own analysis of arguably his most famous paradox of all: The race between Achilles and the tortoise.


Introduction

Zeno, son of Teleutagoras, was born in Elea, Lucania (now southern Italy) around 490 B.C. Zeno was a member of the Eleatic school, and most of what we know personally about him is from Plato's dialogue Parmenides - "tall and fair to look upon", he is thought to have been adopted by Parmenides as his own son (it is even suggested that they may have been lovers).

Parmenides, founder of the Eleatic school, saw the physical world as an elaborate illusion devoid of Truth, for it is constantly in flux and seems to defy what logic might conclude: that nothing can come from nothing, and therefore change cannot come to be. The only thing that can be called Truth must be timeless, uniform, and unchanging; and the only way to this Truth is to reject the illusion of reality and declare all existence as One.

Zeno cleverly defended this point of view though his paradoxes, which attempt to undermine the reality of that which Parmenides saw as illusion. Unfortunately, none of Zeno's writings have survived; in fact, according to Plato, it seems that the paradoxes were not even published by choice. Plato writes:

... a youthful effort, and it was stolen by someone, so that the author had no opportunity of considering whether to publish it or not. Its object was to defend the system of Parmenides by attacking the common conceptions of things.

Even though he dismisses Zeno's paradoxes as flawed, Aristotle himself credits Zeno for inventing the dialectic; a method of argument where apparently contradictory ideas are placed in juxtaposition as a way of establishing truth on both sides, rather than disproving one argument; a form of argument used so successfully by Socrates, and many a philosopher hence.

Aristotle describes Zeno's The Achilles as follows:

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

This description of the paradox belies the nuanced complexity of the race and the way in which it undermines our commonly held notions of motion - which is all the more unhelpful in the context of the modern mind's aversion to both nuance and complexity. To help with understanding of the paradox, let me restate it in terms of a common programmer's triad: input, body, and output; or, more traditionally, assumptions, evaluation, and conclusion (this topology is, I hope, an apt description for the mostly technically-minded of the K5 community):

Achilles and the tortoise decide to have a race. Achilles is known to be the faster runner of the two, and therefore decides to give the tortoise a head start.

Once the race begins, it is true to say that Achilles will take some time to reach the starting point of the tortoise. During this finite time, it is also true to say that the tortoise will have moved forward by some finite distance, and will therefore still maintain a lead in the race. Achilles will once again take some time to reach the tortoise's new position, during which the tortoise will move forward some distance yet again, thus maintaining his lead.

This continues on forever. Therefore Achilles never overtakes the tortoise.

We can see here that the initial conditions establish the relative speed and starting positions of each of the participants (the tortoise is slower-than and starts in-front-of Achilles); the body is a relativistic evaluation of the change in their positions (relativistic in the sense that Achilles' position is evaluated relative-to the tortoise's position, and vice versa); and the conclusion is that Achilles, though the faster runner, will never overtake the tortoise.

And herein lies the paradox: we know Achilles will overtake the tortoise; so why, in the process of evaluating the race, using seemingly sound initial conditions and seemingly correct evaluations, do we come to such a nonsensical conclusion?

A Mathematical Solution

The power of Zeno's fable lies in its absurd conclusion about a most fundamental aspect of existence: motion. From the day we are born, we learn of the world through our instinctive movement within it - suckling on teats, reaching out to the face of the owner of said teat, sticking all manner of objects we come across in our ears, nose, and mouths, etc. We explore our environment through movement, and we explore the environment and behaviour of external objects by moving them about, presenting them to differing environments, and observing the effects of these differentiations (the most popular type of motion being that designed to break said object).

Some of us, in fact, never grow out of this mode of exploration.

Motion is not only fundamental in our experience of the world; it is also the basis of our greatest abstract concept: time. Thus, when a paradox is presented that seems to undermine this most fundamental of instincts, it is easy to dismiss - and one tends not to be too sympathetic to an instrument that, if taken seriously, makes our intellectual life so uncomfortable and makes a mockery of our instinctual attachments to our own reality. A resolution of the paradox requires that we either think out and resolve the intellectual difficulties presented while preserving our instinctual attachments, or we remove these attachments outright and re-evaluate the meaning of the conclusions in terms of the mathematics implied, and accept them without further questioning.

When it comes to attachment-free evaluation, mathematics is our most potent tool. And, indeed, a mathematical solution to Zeno's paradox satisfactorily resolves the conundrum: Zeno makes the fatal assumption that a sum of an infinite amount of terms implies an infinite sum. For Zeno's conclusion to be true, the tortoise must be able to maintain a lead over Achilles over the full distance of the race (in this case undefined, and therefore infinite), but as we add up the potentially-infinite amount of forward movements made by the tortoise, we come to a fundamental limit as to how far he can move forward while still maintaining his lead. And even though we could use an infinite number of terms to calculate this limit, we can definitely say that the limit itself is not infinite - the tortoise simply cannot maintain his lead over all distances of the race.

The mathematician can conclude that Zeno's paradox comes about because of a confusion of infinities: that, simply, an infinite sum does not imply an infinite result.

On Natural Language

Given this satisfactory mathematical solution, it is tempting to declare the paradox solved; after all, shouldn't a mathematical solution suffice for a problem that can be fundamentally reduced to mathematical statements?

To the mathematician, the answer is a hearty "Yes!" - the problem is solved, and need not be considered any further. To the layperson however, the answer is not so clear. "Where", pleads the layperson, "did we resolve the fundamental incompatibility with the true assertion that the evaluation [can] continue forever, and the seemingly sound conclusion that therefore, Achilles never overtakes the tortoise?"

To better illustrate the layperson's reservation, consider the following mathematical statement:

1 + 1 = 2

To a mathematician, this assertion is almost axiomatic; its conclusion is to be accepted and need not - should not - be further questioned. Indeed, the layperson would tend to agree; after all, this statement is often used as a benchmark of truth, even by laypeople themselves! But translate this statement into natural language and this certainty begins to wane:

Adding one thing to another yields two things.

Suddenly, the mathematically axiomatic statement becomes that little bit less certain: what if the "things" to which we refer are "drops of water"? Adding one drop of water to another does not necessarily make two drops of water - "1 [larger] drop of water" is also a valid answer. Or what if we added a prawn ("one thing") into a bucket containing a crab ("another"); will we end up with "two things"? Perhaps, but only for as long as the crab resists the temptation to eat the prawn; once the inevitable happens and the crab succumbs to instinct, we will once again be left with only "one thing."

It is obvious here that some ambiguity has been introduced in the translation of a purely mathematical statement into natural language: mutable objects do no behave as predictably as immutable numbers. Of course, this in no way exposes a fault in the mathematical statement, but by the same token the mathematical statement does not define the behaviour of real-world objects; it is only as accurate as its application and interpretation.

Conversely, a mathematical statement is only as intuitive as its best interpretation; and it is here that the mathematical solution to Zeno's paradox comes up short. Quite simply, the mathematical solution does little to pander to instinct: it simply declares that the paradox is not real, and thus does little to add to its understanding in the layperson's mind.

Aliens and the Art of Boxing

Interdisciplinary language translation (in fact, any form of language translation) is more often than not fraught with subtleties that can be extremely difficult to detect, often requiring a lifetime of immersion to appreciate. Mathematics is a discipline where such translations are as important as the statements in the language itself - absence its isomorphic binding to reality, mathematics is just a bunch of symbols; and not very pretty ones at that.

But there are still instances in mathematics where such translations - interpretations - are difficult to extract. Physicists, for example, have a difficult time interpreting the mathematics of quantum mechanics in terms of the observer-independent reality they purport to represent.

To better illustrate the type of translational subtlety that hinders a better understanding of Zeno's paradox, imagine the following situation: your friendly neighbourhood aliens are about to visit earth, but this time they want to take in some human culture, and so they leave their anal probes and maps of California at home; they want to watch a boxing match.

Unfortunately, centuries of anal probes have failed to extract a meaningful definition of that whacky human behaviour known as sport; it is up to you to explain the art of boxing to the aliens.

There are many ways to explain the sport of boxing, but for brevity let us limit ourselves to two: we can explain it in terms of the evolutionary adaptation of instinctual forces that shaped the male of the species; survival, bravado, and mate acquisition all played a role in the formation of the ideas behind the modern version of the sport we know today. Alternatively, we can explain it in terms of what we think about it: as a hobby, as entertainment, as a fitness regime; as a sport.

The first of these options seems the most appropriate to use in this case: the aliens should be familiar with the mechanics of survival, and the strategies used therein, given that they have presumably used such strategies to beat out their local competition. Talking about boxing in terms of the second option seems to lead to circular arguments: boxing is a sport because it's entertaining; boxing is often a hobby because it's a popular sport; etc. It's difficult to get "traction" in this form of explanation, for it is not based in any concepts that we know the aliens can understand and appreciate.

After a few lessons in the evolution of boxing and, as a natural progression, in human physiology, the aliens may begin to wonder why "below the belt" blows are considered illegal; indeed, a good whack to the goolies, according to the alien's calculations, should easily bring down the largest of opponents. "Should an event that panders to the instinctual urges of survival", pleads the alien, "deny some of the possibilities that a toe-to-toe show of muscellry could theoretically allow?"

Once again we can either continue to explain this behaviour in terms of the evolution of the sport - the subjugation of modern man's survival instincts; the introduction of rules as a means of controlling the environment of these behaviours; the subtle tweaks of these rules over time - or we could explain it in terms of higher-level concepts like entertainment. This time, however, it is the evolutionary explanation that seems to lack "traction" in its ability to enlighten - the rule forbidding "low blows" can be better explained as a means of maximising entertainment by allowing for a more evenly matched contest; prolonging the spectacle, prolonging the opportunity for the consumption of alcohol, prolonging the opportunity for social interaction.

Just as an evolutionary explanation of specific rules of boxing can be unhelpful in attempting to understand the reasons for the rules, a mathematical explanation of Zeno's paradox can be seen as unhelpful when attempting to understand the reasons for the paradox's non-existence.

Zeno's World

Before I begin to expand on a description of Zeno's paradox, let me make an initial statement of principle; a truism that need not be explicitly stated, but can be a helpful tool in understanding of Zeno's fable:

A statement about a World is but merely one aspect of it.

Or, in other words:

The knowledge contained in a statement about a World is less than the amount of knowledge contained in the World itself.

This first principle in dealing with natural language is just a concession that we are dealing with summaries of reality and NOT reality itself; it is merely an admission that the World that Zeno describes in his fable is not only composed of the information given by him; the World he describes is larger than his (simple) description.

This concession is easy to grant - for if it were not true, then we can simply conclude that Zeno's paradox comes about because he describes a world that in no way reflects the behaviour of our own world; that in Zeno's World, it is indeed impossible for Achilles to overtake the tortoise because this is how the World has been defined. In this Fantasy World of Zeno's, the "paradox" is a valid state, and is only considered a paradox because of our instinctual tendency to compare this Fantasy World with our own.

Once we accept this first principle we can immediately ask the following question: does Zeno's World allow Achilles to be in-front-of the tortoise?

If the answer is no, then we're back to Zeno's Fantasy World, and we can dismiss the paradox as having no bearing on our own world.

If, however, the answer is yes, then we can begin to explore the behaviour of Zeno's World in this new state: what happens if we tweak the original description to allow Achilles to be in-front-of the tortoise? Here is a translation of the original description of Zeno's World with one vital difference: the initial conditions have been changed so that Achilles begins the race in front of the tortoise (changes are in bold):

Achilles and the tortoise decide to have a race. Achilles is known to be the faster runner of the two, and therefore decides to give himself a head start.

Once the race begins, it is true to say that the tortoise will take some time to reach the starting point of Achilles. During this finite time, it is also true to say that Achilles will have moved forward by some finite distance, and will therefore still maintain a lead in the race. The tortoise will once again take some time to reach Achilles' new position, during which Achilles will move forward some distance again.

This continues on forever. Therefore the tortoise never overtakes Achilles.

It is important to note that this new description of Zeno's World is exactly the same as the original - it has merely been translated to allow differing initial conditions.

We can now see that the paradox seems to disappear: Achilles does indeed seem to be able to stay in front of the tortoise, forever! But such a conclusion would be premature - where on earth could you have such a race where the evaluation part of this story is true forever? Surely, after travelling approximately the distance of the earth's circumference Achilles will no longer be in-front-of the tortoise, but behind! In fact, where in the universe could one continue such a race forever such that Achilles will never "catch up to" the tortoise, and thus never end up behind him? It is, after all, the opinion of physicists / cosmologists that if you traverse the universe in a straight line, you will eventually end up at the same spot that you started.

Indeed, this new version of Zeno's story is as much a "paradox" as the original: in the original version the conclusion is invalid because in our experience of the world, Achilles will overtake the tortoise; in the new version the conclusion is invalid because in our scientific knowledge of the universe, Achilles will catch up to the tortoise and eventually end up behind him.

In neither version is the statement "this continues on forever" true.

On Structural Duality

This description of the paradox is intuition-friendly because it is dualistic in nature; we expose "both sides of the coin", so to speak, and allow our intuition to compare and contrast the two opposing sides. Duality is a common structural theme in our knowledge of the world; Plato's Ideas, the wave/particle duality of light, the mind/body problem - just to name a few - are defined in terms of a duality ideas; that which is and that which is not.

In fact, a dualistic view of the world seems to be at the core of our ability to understand anything at all; our brains are, after all, composed of two equal hemispheres. It could be true that intuitional meaning is more easily extracted when we are able to consciously split a concept at the highest level, and allow the mechanics of our mind's functional unification to bind these concepts into our naturally-dualistic worldview.

I will admit that this is all just romantic conjecture. Still, one can hardly deny the attraction of such dualistic mysticism - its been part of human culture forever, most pronounced in the form of our dualistic belief system; the dichotomy of heaven and earth is but one example.

In any case, we can now begin to better understand Zeno's fable. Zeno seems to describe only half of Zeno's World, conveniently allowing us to falsely believe that that the alternative state - where Achilles is in-front-of the tortoise - is impossible to attain simply because he offers no easy transition into that state. In reality, Zeno's paradox merely exposes the difficulty in describing this transition from one state to the other, but does not explicitly forbit it.

The question of this transition is still open: when/how does Zeno's World transition from one state to another? While beyond the scope of this article, one could argue that this is still an open question. Physicists may be able to explain the mechanics of this transition in reality (for example, by invoking the lower-limit of motion as a quantum state transition), but explaining how this happens in the mental model of Zeno's World is a little more difficult; after all, one could also ask a mathematician how and/or when the statement "1 + 1" becomes equal to "2."

One could even wonder out loud whether this question is valid at all.

The Accidental Genius

"In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance ..."
Russell

It would be a stretch to name Zeno as the creator of the mathematics that was built on the foundation of his paradoxes, but there can be no doubt that his fables have challenged many a generation of thinkers, and have ultimately withstood the test of time. Even now it would be a brave call to declare the paradoxes resolved. Like the many other seemingly unsolvable puzzles that have exercised human thought throughout history, it would seem the best we can hope for is better understanding rather than full understanding.

"Mathematicians, however, ... realising that Zeno's arguments were fatal to infinitesimals, saw that they could only avoid the difficulties connected with them by once and for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or decreasing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please."
Heath

It is perhaps a testament to the universality of knowledge that in trying to break our common conception of motion and plurality, Zeno managed to break our naďve concept of infinity.

And for that, he will be remembered forever.

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Display: Sort:
Deconstructing Infinity: An Analysis of Zeno's Paradox | 262 comments (207 topical, 55 editorial, 0 hidden)
Good stuff. (1.33 / 3) (#7)
by haplopeart on Wed Jan 05, 2005 at 12:52:49 PM EST

I only hope it survives in the queue.
Bill "Haplo Peart" Dunn
Administrator Epithna.com
http://www.epithna.com

Paradox... (2.72 / 11) (#10)
by Znork on Wed Jan 05, 2005 at 01:19:01 PM EST

Frankly, I've never quite gotten this paradox. The more iterations you perform the shorter the time interval becomes, reaching an infinitely small time interval.

An infinitely small time interval, as in 'time stops', as in 'within the set time', within the so-called paradox.

This does not equal forever.

Therefore Achilles 'never' overtakes the tortoise within the timeframe before he overtakes the tortoise.

I dont quite see the paradox.

I've often wondered... (3.00 / 2) (#11)
by NoMoreNicksLeft on Wed Jan 05, 2005 at 02:11:02 PM EST

If a few of these "paradoxes" weren't born of some defect in ancient greek, the language.

--
Do not look directly into laser with remaining good eye.
[ Parent ]
As in ... (none / 1) (#12)
by zrail on Wed Jan 05, 2005 at 02:26:47 PM EST

A paradox translated into English could have the same meaning as "A horse is a horse, of course of course" would have translated into Ancient Greek? I suppose its possible. However, I think the more plausible explanation would be that Plato was making it all up and was a little off his rocker, to boot.

[ Parent ]
Not in language but in math. (3.00 / 2) (#25)
by porkchop_d_clown on Wed Jan 05, 2005 at 04:34:13 PM EST

Zeno didn't have the mathematical tools to work with the concepts he was exploring.

OTOH - his genius was such that it occurred to him to actually raise the question.

Has anybody seen my clue? I know I had one when I came in here...
[ Parent ]

Agreed (3.00 / 3) (#13)
by toulouse on Wed Jan 05, 2005 at 03:02:04 PM EST

Even my maths teacher at school used to harp on endlessly about about this particular 'paradox'.

It always seemed to me that this 'paradox' is simply a particular form of the rhetorical trick known as 'begging the question', whereby the conditions of the proposition pre-determine the desired outcome. In this case, Xeno's analysis dictates or pre-supposes, rather than concludes, that Achilles will never pass the tortoise. Cheap sophistry and little more.


--
'My god...it's full of blogs.' - ktakki
--


[ Parent ]
Bertrand Russel diagrees with you. (none / 0) (#14)
by Dont Fear The Reaper on Wed Jan 05, 2005 at 03:06:55 PM EST

Fight!

[ Parent ]
Godel disagrees with Russell (none / 0) (#16)
by toulouse on Wed Jan 05, 2005 at 03:17:33 PM EST

First Round K.O. - waste of money.


--
'My god...it's full of blogs.' - ktakki
--


[ Parent ]
Irrelevant (none / 0) (#19)
by Dont Fear The Reaper on Wed Jan 05, 2005 at 03:35:16 PM EST

What does that have to do with the subject of the article, or Russell's quote?

[ Parent ]
My apologies. (none / 1) (#20)
by toulouse on Wed Jan 05, 2005 at 03:39:02 PM EST

I was simply responding to one spurious argumentum ad verecundiam with another.


--
'My god...it's full of blogs.' - ktakki
--


[ Parent ]
So, (3.00 / 2) (#21)
by Dont Fear The Reaper on Wed Jan 05, 2005 at 03:55:20 PM EST

You say the paradox is sophistry.
I indirectly say that Russell said the "sophistry" "made the foundation of a mathematical renaissance."
You say just because Russell said it doesn't make it right.

To which I'll say you're technically right, but I'll believe him over you, especially based on what else I know, and your lack of actual arguments to the contrary.

To write the paradox off as sophisty and not worthy of consideration at best dismisses all the useful things that happened in mathematics because of people's attempts to address it, and at worst obscures the fact there are still some interesting questions that it has at least some relation to.

[ Parent ]

ror! (2.33 / 3) (#15)
by Dont Fear The Reaper on Wed Jan 05, 2005 at 03:16:33 PM EST

Infinitely small time interval, as in time stops, as in cannot be divided further, as in can't be subtracted from, as is the smallest possible number, as in finite, as in not infinite? As my instructor says: "WHY DID YOU STOP! I DIDN'T TELL YOU TO STOP!"

[ Parent ]
Somebody so keen to invoke Russell (none / 0) (#18)
by toulouse on Wed Jan 05, 2005 at 03:27:31 PM EST

should at least have engaged the original Principia before putting their mouth into gear. Performing mathematical calculations with infinitely small quantities is commonplace, and underlines most of modern physics.


--
'My god...it's full of blogs.' - ktakki
--


[ Parent ]
Someone should remember what a limit is (none / 0) (#114)
by curien on Fri Jan 07, 2005 at 02:06:47 AM EST

It's not "performing mathematics with infinitely small quantities." It's observing a trend in the results of calculations with increasingly small numbers, such that one may observe what the result of the calculation approaches (but never actually reaches) as some numbers approach (but never actually reach) zero.

We write calculus without the baggage of "lim" all over the place, but "d/dx" is just an abstraction of it.

--
This sig is umop apisdn.
[ Parent ]

That's because you understand (3.00 / 3) (#24)
by porkchop_d_clown on Wed Jan 05, 2005 at 04:26:46 PM EST

infinitisimals. The concepts of infinity and infinitisimals were not really understood way back then; what Zeno was describing couldn't really be expressed mathematically till limits and integrals were developed.

Has anybody seen my clue? I know I had one when I came in here...
[ Parent ]
Not always so (none / 1) (#115)
by curien on Fri Jan 07, 2005 at 02:17:30 AM EST

According to your reasoning, any sequence of numbers that approaches zero as you approach the end of the sequence would have partial sums that converge to a finite number, but this is not the case.

For example, consider the sequence 1, 1/2, 1/3, 1/4, ... , 1/n. The sum of the elements of this sequence

        n  
       --
       \  (1/k)
       /
       --
       k=1

increases without bound as n->inf. The fact that 1/k becomes "infinitessimally small" and approaches zero matters not.

--
This sig is umop apisdn.
[ Parent ]

Some yes, some no (none / 0) (#253)
by ernest on Sat Jan 29, 2005 at 05:25:46 PM EST

I don't think anybody wanted to claim all such sequence ended somewhere finit. It's just that Zeno's paradox assume this to be the case for the tortoise's movements.

[ Parent ]
paradox shmaradox (2.33 / 6) (#23)
by j1mmy on Wed Jan 05, 2005 at 04:19:06 PM EST

What's the point of trying to reason with a talking animal?


Achilles was an animal? (none / 0) (#41)
by gdanjo on Wed Jan 05, 2005 at 08:14:04 PM EST

Dan ...
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]
The paradox is an illusion (2.82 / 17) (#26)
by jd on Wed Jan 05, 2005 at 04:36:32 PM EST

It's so much simpler than your explanation. However, the solution was not to be mathematically established until Newton/Descartes laid down the principles of calculus.

Calculus is where the paradox collapses. In the same way that, when you take the difference of two points in an equation to the limits, the gradient does NOT become infinite or zero, but tends to the limit that defines that equation, when you reduce the timeframe in the race to zero, the difference does NOT collapse but tends to the limit that defines the equations describing the change in relative speed.

When you divide by zero, you get a nonsensical result. But when you differentiate, you get something that is valid, even though (in a sense) you are doing exactly the same thing.

Zeno did not have a copy of the Principea in front of him, but if he did, he would have rapidly understood why the paradox doesn't hold. It is all to do with how you handle the zeros.

There are far better paradoxes to study (such as the Library Paradox) which are less amenable to trivial analysis. Many so-called paradoxes (such as the Liar's Paradox, Olber's Paradox, etc) are purely a fault of the way in which you treat the numbers. The problem itself, if correctly examined, is quite trivial.

sure (3.00 / 2) (#34)
by gdanjo on Wed Jan 05, 2005 at 07:38:35 PM EST

to the mathematician, the paradox does not exist - I thought I made that clear. This is an attempt at an intuitional explanation.

Without regular immersion in maths, it's difficult to understand why the fable fails to be a paradox - conversely, when one is fluent in maths, any other type of explanation is long, complex, and unweildy.

I always knew K5 would be a tough audience to convince that a non-mathematical "solution" is useful, given that most K5ers, I would guess, are quite comfortable in the language of maths. But hey, what can I say ... I'm a masochist :-)

Dan ...
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]

In which case... (none / 0) (#152)
by jd on Fri Jan 07, 2005 at 10:15:08 PM EST

Masochist: Hurt me!
Sadist: No!

[ Parent ]
I think you underestimate the paradox (2.75 / 8) (#38)
by speek on Wed Jan 05, 2005 at 07:56:24 PM EST

I used to think the answer was Calculus too, but your explanation doesn't help, I don't think. Zeno is not dividing anything by zero, but he is assuming an infinitely divisible reality and infinitely divisible time. In other words, fully analog reality.

The idea of asymptotically approaching a point actually strengthens the paradox, IMO. No matter how close the hare gets to the tortoise, he is still behind it, and the tortoise still gets to move some infinitesmal distance further. To clear the paradox as you would like, we'd have to assume that there existed some measure of time during which the hare moved and the tortoise did not. And there seems to be no such time period in a completely analog system.

In a digital, or "quantized" system, one could find a time period during which the tortoise failed to "jump" ahead to the next available space, while the hare did jump ahead and drew even with the tortoise. Which, to me, sounds a lot like what the Heath quote was talking about, in the article.

Another potential resolutions of the paradox, that I can think of, are accepting the lesson from the theory of relativity that time is not at all what the way we intuitively think it is, and so maybe time may pass for the hare but not at all for the tortoise, thus letting the hare catch up.

And, still another potential resolution is a Bohm like reality (Implicate Order) in which motion is in fact not real and is an illusion.

All these resolutions dramatically change our natural perception and assumptions about the world, however (ie reality is quantized, time is not as we see it, or motion is in fact, an illusion).

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

You overestimate it. (3.00 / 3) (#55)
by mcc on Thu Jan 06, 2005 at 12:46:33 AM EST

The paradox is in a sense tautological-- it presents a model of reality and then demonstrates that model to present seemingly nonsensical consequences. You do the same thing. Unfortunately this doesn't say anything about reality-- only your model. And what it says is that the model is poorly designed.

If you have to phrase the question this carefully in order to make it confusing, then maybe the only real problem here is that you're just phrasing it poorly.

[ Parent ]

hmm (none / 1) (#59)
by gdanjo on Thu Jan 06, 2005 at 03:12:10 AM EST

The paradox is in a sense tautological-- it presents a model of reality and then demonstrates that model to present seemingly nonsensical consequences.
I agree, and I thought I made this point clear in the story - where I say that Zeno's World is only paradoxical if we are "tricked" into thinking the model presented reflects our own reality. I also show that to not get tricked, you need to think of the story in terms of Achilles begining in front of the tortoise - in this state, it becomes obvious that "Achilles never overtakes the tortoise" is true, and is no longer a paradox.

The trick is to get this conclusion to "click" in the non-mathematically inclined mind, without resorting to the non-mathematicians "mystical" belief in mathematics, and without requiring them to fully understand mathematical concepts.

Unfortunately this doesn't say anything about reality-- only your model.
Every statement is a "model" - I'm not sure what you consider a statement about reality, and how you distinguish this from a statement about a model. An example would be nice.

Dan ...
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]

ugh (none / 0) (#60)
by gdanjo on Thu Jan 06, 2005 at 03:14:19 AM EST

obviously, I meant:

it becomes obvious that "Achilles never overtakes the tortoise" is true

to be:

it becomes obvious that "Achilles never overtakes the tortoise" is false

Dan ...
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]

Well (none / 0) (#63)
by mcc on Thu Jan 06, 2005 at 05:20:00 AM EST

I'm not sure what you consider a statement about reality, and how you distinguish this from a statement about a model.

I was mainly responding here to speek, and his attempts to maintain the paradox's relevance in the face of a simple way to avoid it.

As far as what I would consider a statement about "reality", I was trying to indicate that certain statements about models can be used to demonstrate more general statements that can be said to have some useful application even outside of the model. Goedel's incompleteness theorem, for a random example, demonstrates implications that you have to contend with even if you don't share the particular model of logic used to prove the theorem. The point I was trying to make was that Zeno's paradox doesn't exist at that level of generality. It's avoidable; you don't really have to deal with it if you don't want to.

[ Parent ]

we're always only talking about models (none / 0) (#69)
by speek on Thu Jan 06, 2005 at 07:38:28 AM EST

Like I said, Zeno assumes an infinitely divisible reality (that's his model), and then demonstrates a problem with that model. Do we take it as given these days that reality is not infinitely divisible?

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

No. (none / 0) (#151)
by jd on Fri Jan 07, 2005 at 10:13:04 PM EST

Matter is quantized. Energy states for electrons are quantized. Beyond that, there's nothing that is solidly known. It is a prediction of QM that everything is quantized, but QM is incomplete and is incompatiable with Relativity. It is therefore possible that QM has flawed assumptions built into it.

(Unlikely, because QM holds up very well, but possible, because Relativity ALSO holds up very well.)

[ Parent ]

what are you people, the wikipedia generation? (2.80 / 10) (#62)
by SIGNOR SPAGHETTI on Thu Jan 06, 2005 at 04:51:07 AM EST

the limit argument, which merely describes the mathematical content of zeno's argument in modern mathematics, does not resolve its physics paradox. it concedes it. zeno's is a paradox of physics that depends on the notion space and time are continuous and infinitely divisible. they are not. special relatively has taught us zeno was correct: motion is impossible under the assumptions of classical mechanics, assumptions you reiterate. the problem with Introduction to Calculus is it's not physics.

--
Stop dreaming and finish your spaghetti.
[ Parent ]

Special Relativity (none / 1) (#150)
by jd on Fri Jan 07, 2005 at 10:07:54 PM EST

...had nothing to do with discrete units of time OR space. That's Quantum Mechanics, which has not yet been scaled to anything on the macrocosmic level, such as the physically-observable Universe.

(There isn't a computer large enough or fast enough to handle the QM equations for a complex molecule, never mind a complex planetary system.)

Further, QM and Relativity conflict. They cannot both be true. So far, nobody has established which is the one that is flawed, largely because nobody has figured out how to get them to apply to the same system, but also because QM is still too incomplete. There's no QM model for gravity, for example, so you can't create a scenario involving gravity in which the two theories produce different predictions.

There is no evidence of any kind that either space or time are quantized in any way. I expect them to be, because space cannot be empty and mass IS quantized. However, this is not proof. This is merely an expectation. There is no proof.

Back to the paradox - it has nothing to do with the idea that space or time are continuous. It has to do with the fact that the measurements are asymtotic to something other than zero. Space and time can both be continuous, and the observations would remain the same.

[ Parent ]

what? (none / 1) (#159)
by SIGNOR SPAGHETTI on Sat Jan 08, 2005 at 01:35:22 AM EST

Friend, your hand waving is incoherent. I'm sorry. I had to read your comment twice and I'm still not entirely convinced you didn't reply to me by mistake because QM has nothing to do with anything I wrote.

Here's an object o: o

Look at it. Memorize the way it looks.

Here's a picture of the SAME object in the context of continuous motion:

... oooooooooooooooooooooooooooo ...

See it moving? There are more than seven fillion dillion instances of o in that picture. There are so many we can't count them.

PICK ONE.

That one? OK, let me ask you something. The one you picked, does it look different than o at rest? Does it look different than any of its seven fillion dillion snapshots in motion? At any so-called instant, does it have some magic quality that transmits causally the fact it is or isn't in "motion?"

No. Ergo, motion is an illusion.

Calculus is where the paradox collapses. In the same way that, when you take the difference of two points in an equation to the limits, the gradient does NOT become infinite or zero, but tends to the limit that defines that equation, when you reduce the timeframe in the race to zero, the difference does NOT collapse but tends to the limit that defines the equations describing the change in relative speed.

What you're struggling to describe above using words in the language of continuous functions (for their je ne sais pas quoi air of mathematical legitimacy, I guess) is "instantaneous velocity," or dx/dt. In this model of motion, objects occupy space -- absolute space -- and propagate in a sequence of "instants" through time -- absolute time. If one wants to get from point a to point b, one boards the magic function and gets off at the designated asymptote. Or something equally preposterous believed by atheists who scoff the concept of God. I find you have to get up very early to pull the wool over their eyes, let me tell you.

Anyway, if you accept this model then you concede the paradox because there is no way to say, when o is moving, that it knows it's moving. That's the point of the paradox. It's a paradox of physics, not mathematical reasoning.

Except there is a way. For it turns out that o in motion occupies a different plane of simultaneity, with the usual relativistic consequences. Each one of the seven fillion dillion snapshots of o in (relative) motion IS different. It looks different to an observer at rest, and the observer to it.

--
Stop dreaming and finish your spaghetti.
[ Parent ]

Motion, et al (none / 1) (#171)
by jd on Sat Jan 08, 2005 at 07:32:15 PM EST

  1. If motion were an illusion, the Heisenberg Uncertainty Principle would not apply. Since it does, motion is not.
  2. Quantization (discrete time & space) is a property that can be derived solely of QM, not of Relativity. Ergo, either you were spouting BS about relativity, or you were referring to QM. My point was that which was irrelevent.
  3. Yes, you can distinguish between any one "o" and any other "o". You can distinguish by Relativity (which prohibits two events occuring simultaneously, and therefore prohibits two objects being equidistant from the observer, thus allowing you to tell them apart). You can also distingush them by QM, because the probability waves defining the object are different.

If you can't understand the post, don't blame it on incoherence of the poster, unless you've firm proof you've at LEAST equal understanding of the subject. Your posting about 'o''s proves you do not. Sorry.

[ Parent ]

let's start small (none / 0) (#176)
by SIGNOR SPAGHETTI on Sat Jan 08, 2005 at 11:28:49 PM EST

here's a lemon. perhaps you'd like to make invisible ink?

--
Stop dreaming and finish your spaghetti.
[ Parent ]

Lemon juice as invisible ink (none / 0) (#183)
by jd on Sun Jan 09, 2005 at 04:59:01 PM EST

That was very popular with the Catholics, when the Protestants were ruling England, as at that time, it was largely a Catholic secret. Lime juice is supposed to work better, though, although any citric fruit will work.

Stealth technology has produced invisible inks that work the other way, too, becoming invisible over time. These work using something that sublimes at room temperature. Iodine crystals are good for this. Under normal conditions, iodone is not stable at room temperature and pressure in solid form. It sublimes, becoming 100% gaseous, without going through a liquid state. Inks based from similar materials are only "stable" for a period of time, before vanishing.

These days, using a pen with such an ink is a problem, as you can identify depressions in the paper using very finely-ground carbon or iron particles, with a magnetic field on the other side of the paper. Anything loose can be displaced, showing even the faintest depression. This was used with great effect to free many wrongfully-convincted prisoners in Britain after a corruption scandal at the West Midland's Serious Crime Squad.

On the other hand, an inkjet should be able to use either of the two classes of ink described above, leaving no tell-tale depressions.

[ Parent ]

i did not know that (none / 0) (#184)
by SIGNOR SPAGHETTI on Sun Jan 09, 2005 at 05:07:26 PM EST

that's ok, i'm not ashamed i don't know everything. i'm proud of knowing i know so little, actually.

--
Stop dreaming and finish your spaghetti.
[ Parent ]

Pride and knowledge (none / 0) (#199)
by jd on Mon Jan 10, 2005 at 07:43:17 PM EST

If all of human knowledge, throughout all of history and pre-history, were to be represented by the grains of sand in a child's sandpit, then the sum total of all that is knowable but unknown is greater than the sum of all the sand on Earth and Mars combined, and the total of all that is unknowable exceeds the knowable by as much as the knowable exceeds the known.

Take any human being, however "bright", however knowledgable, and compare that knowledge to all that humanity has ever known, that person would pale into utter insignificance. Compared, then, to all that is knowable, or all that is true but unknowable, a person is nothing.

I tell you then to delight in what you DO know, great or small. Pride is taking the light by which you see the world and sheltering it so much that even you cannot see by it. In the same way, no serious scientist can EVER hope to understand anything, if they lose their childlike wonder, no human being can EVER hope to do anything with that understanding if they lose their own childlike delight in the new.

Love not the darkness when you can enjoy the stars. You cannot see the stars without the darkness, but neither can you see the stars if all you look at is the darkness.

[ Parent ]

Er. (1.50 / 10) (#27)
by ubernostrum on Wed Jan 05, 2005 at 04:47:39 PM EST

I don't think you really understand the point of the paradox...




--
You cooin' with my bird?
What makes you say that? (n/t) (none / 0) (#170)
by The Archpadre on Sat Jan 08, 2005 at 04:53:22 PM EST


__
Where did my waffles go?


[ Parent ]
Oh, I don't know... (none / 0) (#175)
by ubernostrum on Sat Jan 08, 2005 at 10:21:08 PM EST

The fact that Zeno's paradoxes are really only meant to show that the Pythagorean concept of space and time leads to absurd conclusions?




--
You cooin' with my bird?
[ Parent ]
-1 how and why (1.14 / 14) (#45)
by Lady Writer On The TV on Wed Jan 05, 2005 at 08:40:45 PM EST

did you manage to write scores of paragraphs for a phenomena that can succinctly and simply be understood in one or two sentences?

i didn't read it, but there's either a lot of filler, is pseudointellectual, or is wrong. any case,  its -1

pinggggggg (1.80 / 5) (#51)
by Your Moms Cock on Wed Jan 05, 2005 at 09:25:07 PM EST




--
Mountain Dew cans. Cat hair. Comic book posters. Living with the folks. Are these our future leaders, our intellectual supermen?

[ Parent ]
ponggggggg (3.00 / 3) (#68)
by noogie on Thu Jan 06, 2005 at 07:33:03 AM EST




*** ANONYMIZED BY THE EVIL KUROFIVEHIN MILITARY JUNTA ***
[ Parent ]
wow an immediate -1 (1.00 / 22) (#50)
by Your Moms Cock on Wed Jan 05, 2005 at 09:23:01 PM EST

no second thoughts about that one lol


--
Mountain Dew cans. Cat hair. Comic book posters. Living with the folks. Are these our future leaders, our intellectual supermen?

This is what makes philosophy so fun: (2.66 / 9) (#52)
by JChen on Wed Jan 05, 2005 at 10:02:50 PM EST

The more you doubt it, the more you want a concrete answer. Yet everything that we have to explain our natural world is merely a set of tools used to describe what we perceive; calculus is not the truth- it is a metaphor for what we perceive to be true, and a pretty good and practical one at that.

However, it scares me when people accept it as is, that there is a limit and subsequently dismissing this problem afterwards as if it is solved; no, I think it is merely one way of saying that it is solved. It is satisfactory to our judgment, but that does not mean it is the truth as it truly exists, if it even does.

I think it's not really a question of solving the paradox itself, but the methodology of how one approaches such a problem; don't obsess over not being able to find the Truth, but don't dismiss it as solved either.

Let us do as we say.

The Truth? YOU CAN'T HANDLE THE TRUTH (none / 0) (#215)
by genjilad on Thu Jan 13, 2005 at 04:33:16 AM EST

Nobody can. Or they can, but only their own little personal Truths, through the lenses of those philosophical beer goggles that we see the world through. One goth guy sees a foxy lady, a jock sees some fat goth chick, and a cannibal sees dinner for the next three weeks. Where's the objectivity in that? Huh? See? Philosophy IS fun

[ Parent ]
space & time are not scale invariant (2.20 / 5) (#58)
by SIGNOR SPAGHETTI on Thu Jan 06, 2005 at 03:11:03 AM EST

your metaphysics are old school. damn trolls. lol your linked articles are gibberish.

--
Stop dreaming and finish your spaghetti.

Wow (2.77 / 18) (#61)
by felixrayman on Thu Jan 06, 2005 at 03:47:17 AM EST

That's a rather long-winded way of telling us all that you have no fucking clue what an integral is, dontcha think? There was an excuse for that in 490 B.C., what's yours? Took "Business Math" instead of calc (or whatever term they use nowadays for the math class that includes a bonus tour of the local box factory, in which you will someday, if all goes well, be employed)?

Call Donald Rumsfeld and tell him our sorry asses are ready to go home. Tell him to come spend a night in our building. - Pfc. Matthew C. O'Dell

I salute you, sir. -nt (none / 0) (#78)
by Kasreyn on Thu Jan 06, 2005 at 11:43:57 AM EST

nt
"Extenuating circumstance to be mentioned on Judgement Day:
We never asked to be born in the first place."

R.I.P. Kurt. You will be missed.
[ Parent ]
That's a rather clever way (3.00 / 2) (#82)
by gdanjo on Thu Jan 06, 2005 at 04:32:59 PM EST

of telling us you didn't read the article:

And, indeed, a mathematical solution to Zeno's paradox satisfactorily resolves the conundrum: Zeno makes the fatal assumption that a sum of an infinite amount of terms implies an infinite sum.
But kudos, sir, for the most creative insult yet (though, Simpson references are so 1990's - you work on that and I'm sure your wookies will stop bending themselves!).

Dan ...
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]

Doh! (none / 0) (#112)
by felixrayman on Fri Jan 07, 2005 at 01:44:34 AM EST

Actually when I took the class, it was a shirt factory. But since all those jobs long ago moved to Guatamala or the Mao/Nike Garment Correctional Facility or where the hell ever, I tried to slip the Simpsons reference in so it wouldn't look so 80s.

You busted me.

I did learn how to balance a checkbook in the class, but I think my fellow students may have been better off learning how to fill out a payday loan application, or learning how to estimate ones chances of winning the lottery given that one spends 30% of a weekly minimum wage paycheck on tickets. (A: practically guaranteed!).

Call Donald Rumsfeld and tell him our sorry asses are ready to go home. Tell him to come spend a night in our building. - Pfc. Matthew C. O'Dell

[ Parent ]

Doesn't solve anything. (3.00 / 3) (#64)
by gyan on Thu Jan 06, 2005 at 06:07:06 AM EST

The mathematical solution is simply a restatement of the paradox in a different framework.

The mathematician can conclude that Zeno's paradox comes about because of a confusion of infinities: that, simply, an infinite sum does not imply an infinite result.

And why do I hold the mathematics as any more 'true'? Do I accept that this result makes sense because it is a mathematical deduction?

 Experientally, I know that motion exists. The paradox comes intellectually. Mathematics is a system of proofs and formal statements created by manipulating axioms (self-evident "truths") using rigid operators. The mathematician's results only appeal to me if I accept the self-evident truths behind them. But if I have doubts about the basis of those 'truths' or their "true" nature, then a deduced mathematics statement, still being human activity, is no more assuring. After all, the direct sensory confirmation is a stronger 'proof'.

********************************

But you "know"... (none / 0) (#126)
by Eccles on Fri Jan 07, 2005 at 10:42:10 AM EST

Your intuition would also tell you that an infinite sum can be finite. (It would be wrong, but good enough for our purposes.) Take a slice of cake. Slice it in half. Slice again. Keep slicing. You get down to crumbs, but the limit on your slicing is your vision, not the cake; intuitively, you would think you could divide that one finite piece of cake into an infinite number of vanishingly small pieces, and that if you add those pieces back together, you get a single piece of cake. So the infinite sum not being an infinite result is, I think, intuitable.

[ Parent ]
but (none / 0) (#138)
by speek on Fri Jan 07, 2005 at 06:18:24 PM EST

When am I going to get around to adding them back together again? After I've "infinitely" divided it? And then how long will it take to add it all up again? I'll let you know when I'm done...

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

I'll tell you why Achilles never caught up (1.80 / 5) (#66)
by tonyenkiducx on Thu Jan 06, 2005 at 07:09:13 AM EST

Because Zeno spent most of his time smoking the ancient Greek equivalent of crack, and never got around to organising the race. This is another example of philosophy(Of which I admit to knowing nothing) self-elevating itself to a higher place, by creating a paradox out of nothing. Dont get me wrong, I know that philosphers have served a purpose in the past, and that they should probably be involved in the modern quest for theories on the creation of the universe, but this is just nonsense from beginning to end.

I have another great paradox for you though. The paradox is in the reasoning, even though it is possible to answer correctly. Four philosophers are stuck in a lift and one farts. Three of them deny it, but the other one points to one of the others and with all certainty says it was him. He reasons to the other Philosophers why that is, and they all believe him.

How could he possibly know this?

Tony.
I see a planet where love is foremost, where war is none existant. A planet of peace, and a planet of understanding. I see a planet called
And your point is? (none / 0) (#139)
by anthroporraistes on Fri Jan 07, 2005 at 06:19:44 PM EST

Actually if the paradox is invalidated (which I'm not going to accept from the context of the article, being that don't think that math is some magic cure all, being nothing more than another philosophical/logical system, complete with external assumptions and disjuncts with reality) it still served a valuable epistemic function.  That is, it shows us a disjunct between external reality, and reasoning.  And it serves as a good point for arguement, as we are doing now.  Though I think the Cretan paradox is better, since it allows discussions (ala Hopstadter) into recursion, and such.  

The creation of the universe is not the domain of philosophy, so why would they want to?  Philosophers are not physicists.  And that is besides the point, since we're discussion a 2000+ year old construct, which does not show where modern philosophy is.  Busy working on logic, epistomology, and ethics, as they should be.  

---
biology is destiny
[ Parent ]

Philosophers used to be involved with science (none / 0) (#181)
by tonyenkiducx on Sun Jan 09, 2005 at 08:28:44 AM EST

And quite a few modern theorists argue that they should be involved again, as a complete seperation of the two fields has occured. Stephen Hawkings is probably one of the most popular names I can think off that has said this. Modern science is advancing quickly in the field of physics, but there are big gaps where a philosophical mind could come up with ideas that might not occur to a physicists mathematical mind. Just my two cents, like I say, I look at this as an outsider, I know naught of Philosophy.

Tony.
I see a planet where love is foremost, where war is none existant. A planet of peace, and a planet of understanding. I see a planet called
[ Parent ]
Because HE was the one who farted (none / 0) (#214)
by genjilad on Thu Jan 13, 2005 at 04:26:16 AM EST

AND HE WAS TRYING TO COVER IT UP

[ Parent ]
Your version of the paradox doesn't follow (3.00 / 5) (#67)
by nusuth on Thu Jan 06, 2005 at 07:25:34 AM EST

This continues on forever. Therefore Achilles never overtakes the tortoise.

These bold words are statements about time it takes Achilles to overtake tortoise. There is nothing in the model that establishes the truth of those statements. The initial sentences only mention time intervals; what happens when Achilles and the tortoise spend a finite amount time. It says nothing about the sum of those finite time intervals.

You may think you have already handled this case by a mathematical solution that proves that sum of infinite number of finite intervals are in fact, in this particular case, finite but that is beside the point. Had Zeno said, "Achilles needs an infinite amount of catching up steps, where each step require finite, non-zero amount of time. The steps collectively require an infinite amount of time, therefore this continues on forever and Achilles never catches the tortoise" paradox still wouldn't follow because it never establishes the truth of "the steps collectively require an infinite amount of time." In the form you stated it, it is no more paradoxical than "I cleverly sneaked anal probes in to the article, therefore Achilles can never overtake the tortoise."

When you add the intuitonal but unstated "no infinite sum can be finite" assumption to the model, the paradox, as stated, follows but can be shown not to exist. As that assumption is a statement established with mathematical terms ONLY, mathematics can prove or disprove correctness of the assumption without refering to any concepts outside of its own domain. There are no crabs, no water drops; just "infinite", "finite" and "sum." And it proves the falsehood of the statement easily.

I don't know if it is possible to restate the paradox in a way that it both follows and exists. Since it can be shown experimentally that motion is possible in this world, a version that both follows and exists must use a world model that is incompatible with our world, therefore its stated assumptions must be wrong in physical domain, not logical domain. You could write this article with that version but this version isn't it.

I'll try (3.00 / 4) (#71)
by speek on Thu Jan 06, 2005 at 07:57:16 AM EST

Trying in terms of existential logic:

We agree that at any point in time prior - no matter how infinitesmally short - to the hare drawing exactly even with the tortoise, the hare is behind the tortoise. But, the hare does draw even at some mathematical point in time (as in, a zero-dimensional point on the timeline).

So, there must exist a time interval as arbitarily short as I want to make it, during which the hare moves from being behind the tortoise to drawing even with the tortoise.

And then I ask you to demonstrate such a time interval, and you say, "easy, at one second before the "even point", the hare is X distance behind. During that one second, he moves and draws even. At which point I say, "fine, but if I cut the second in half, he has not drawn even. Show me the tiniest interval during which the drawing even happens."

etc.

Of course, you can't show me the smallest interval because I can always divide whatever you pick by 2. So you say, "I'll just take the limit, and the size of the intervals approaches zero", which now I can't divide that by two simply because calculus doesn't allow such an operation - it's nonsensical. However, I refer you to this comment which is kind of funny, but really puts a fine point on the problem, I think (as do some mathematicians, apparently).

Part of the point is that it's fine and dandy to write the mathematical notation for a limit => 0, but where's the number? Do we think reality calculates using symbols, or is it just there, "for real", in which there has to be a real measurable number at every step the hare goes through, and none of the hare's intervals get's literally measured as "the limit as interval goes to zero". That would look kind of weird on my infinitely powerful measuring stick.

In some ways, modern physics has answers to this (ie quantum - reality is not fully continuous, relativity - time is not what you think, etc), but all those answers seriously derail our normal intuitions about reality, as does Zeno's Paradox.

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

That is not exactly the same paradox, is it? (1.50 / 2) (#74)
by nusuth on Thu Jan 06, 2005 at 09:56:11 AM EST

You are basically getting rid of the hare and the tortoise; at some point hare overtakes tortoise, but hare cannot reach that point. Which is equivalent to saying moving any distance at all is impossible, restating one other(?) paradox. Let me rephrase your version by using neither time nor velocity:

Moving a distance requires two actions: traversing first half of it first and traversing second half of it after that. As each "travel half of the length" problem is exactly equivalent to the original problem (going from somewhere to somewhere else), it should require no more nor fewer actions to carry out.

So 2*action_count=action_count. So the number of actions to perform is either 0 (eg. we are already at where we want to go) or infinity. In the first case motion is impossible, because we are already at where we want to go. In the second case motion is impossible, because we need to perform an infinite amount of actions to get anywhere.

Now we just need to prove that an unbounded number of actions can be performed in a finite time, for a suitable definition of action, to get rid of the paradox. I've tried that but I'm not sure I'm successful. I can visualize a line segment in a finite amount of time, I can also visualize a point. But imagining a line segment is equivalent to visualizing an two line segments, which when connected end to end is the same single line segment as the original one. In the same way, visualizing a line segment is equivalent to visualizing an infinite number of line segments. Visualizing something is a mental action. Therefore I can perform an infinite number of actions in a finite amount of time, for some suitable actions. Therefore performing an infinite number of actions is possible for some actions.

[ Parent ]

Zeno's paradoxes all are same at root (none / 0) (#91)
by speek on Thu Jan 06, 2005 at 06:08:30 PM EST

My restatement pulls out what I think is the crux of the matter - an inability to comprehend infinity, or infinite recursion, or infinitely small intervals, etc. And what does that mean? Does it mean the infinite cannot exist, or does it just mean our ability to comprehend isn't up to the task?

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

infinity (none / 0) (#100)
by nusuth on Thu Jan 06, 2005 at 06:55:56 PM EST

In which way do you want us to be able to comprehend it? How do you comprehend -1,pi,i,4/27 or 2765646614437712437? In what sense these numbers' existance is different from infinity of non-standart analysis? (I agree that infinities in standard calculus is a bit hairy. They shouldn't be there. And they aren't either. Yet, everybody diverge to them in very much the same way they converge to numbers.)

On an intuitive level, I can comprehend neither of those numbers. 1,2,3 up to (perhaps) 12 is OK. But WTF is 4/27 and how come digits of ratio of circumference to diameter already has my complete biography before I'm dead, with all possible variations, encoded in utf-8? How come my life story (and "Complete Works of Galois" had he dies 50 years after he did) also happens to be logarithm of -1, divided by square root of -1?

Yet, I can use them to understand something else. As tools, I understand them perfectly. I don't think there is any more reality to any of them, small counting numbers included.

[ Parent ]

no reality to the numbers (none / 0) (#123)
by speek on Fri Jan 07, 2005 at 08:19:01 AM EST

Ok, some might say much the same thing by calling our everyday perception of reality an 'illusion'. I can "see" 2 apples, or 5 stars, no? I can represent fractions with real objects, so, on the surface, the simple numbers do seem to have a strong relation to reality.

But you and Zeno appear to think otherwise.

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

2 is in the eye of the beholder (1.50 / 2) (#166)
by nusuth on Sat Jan 08, 2005 at 12:14:29 PM EST

I can "see" 2 apples, or 5 stars, no?

When you see two apples, you construct a collection and having two items is an attribute of that collection. We can discuss for ever whether having two items is an instrinct property of the collection or an agreed upon convention but, fortunately, we don't have to do that. The collection itself is a mental construct, without any reality to it except as a tool for cognizing things. You can look at three apples, and group them into 3 collections of 1 apples each, 2 collections with 2 and 1 apple(s) or a single collection of 3 apples. You can also think of them as a group of four appples missing an apple (eg. when you bought four apples at the market but found only three in the shopping bag when you came home.) Each of the groups' elements exist independently of your mind (eg. the apple doesn't disappear when you don't think about it - unless you are a positivist, then they do disappear) but any collection of them have no reality outside your mind. When you stop thinking about it being that way, the collection and its attributes cease to exist.

I can represent fractions with real objects, so, on the surface, the simple numbers do seem to have a strong relation to reality.

They do. Their relation is a very useful and strong one. However they are not things themselves, they are about things. In fact, their relation isn't even first order (eg. "red" attribute of a red apple); counting numbers are mental constructs about mental constructs.

[ Parent ]

On a second thought, you version isn't paradoxical (1.50 / 2) (#80)
by nusuth on Thu Jan 06, 2005 at 11:59:10 AM EST

...at all.

So, there must exist a time interval as arbitarily short as I want to make it, during which the hare moves from being behind the tortoise to drawing even with the tortoise.

And then I ask you to demonstrate such a time interval, and

...I say, "easy, at time t, runner of the day catches the tortoise. The interval from t-e to t+e is your interval. You chose whatever non-zero e you please." This interval has all tortoise behind hare, hare behind tortoise and the even point. Which also happens to be very close to definition of limit in the modern sense (eg. without calling infinitesmalls to rescue.)

If you want your interval to have only hare approaching tortoise, you will have exactly that. But you would not have a paradox as hare isn't expected to catch tortoise in that interval anyway.

Even when you want to your interval to have both hare approaching tortoise and catching it, but not any moment after that, there is no paradox. As long as we stick to intervals, all is fine. The only problematic case I can see is when you want me to define the last moment hare is still trailing the tortoise. I will fail to do that. But I also fail to see how my inability to do that proves something about motion, which happens during intervals, not moments.


[ Parent ]

moment is an interval (none / 0) (#93)
by speek on Thu Jan 06, 2005 at 06:19:06 PM EST

Yes, I'm asking you to define the last interval during which the hare catches the tortoise, and all I'm saying is that I can still cut that interval in half and leave the hare behind. Isn't that really the point of the paradox, that I can always divide by two? The arrow paradox is the same exact paradox (except dealing more with space than with time), and basically boils down to saying how can an object move through an infinite number of points in finite time? Or vice-versa, how can an object go through an infinite number of time intervals while moving a finite distance?

Hmm, when I put it like that, it looks more like set theory and comparing the cardinality of infinite sets.

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

Just wanted to say (none / 0) (#128)
by sab39 on Fri Jan 07, 2005 at 11:31:36 AM EST

... that that's an ingenious typo for "Infinitesimals" :)
--
"Forty-two" -- Deep Thought
"Quinze" -- Amélie

[ Parent ]
Duh! (none / 0) (#165)
by nusuth on Sat Jan 08, 2005 at 11:46:15 AM EST

As my comment history clearly shows, English is not my strong point. In my native tounge, we literally call them "infinite-smalls." Guess I had never noticed the English word does not follow the same pattern.

[ Parent ]
I don't think he meant it as an insult (none / 0) (#194)
by curien on Mon Jan 10, 2005 at 05:14:55 AM EST

You made one word: "infinitesmalls", which, if sounded out carefully, sounds like "infinitesimals" but looks like "infinite smalls" which, incidentally, is exactly what "infinitesimals" are. It's very clever (even if it was unintentional).

--
This sig is umop apisdn.
[ Parent ]
It wasn't an insult (none / 0) (#195)
by sab39 on Mon Jan 10, 2005 at 01:16:45 PM EST

I wasn't sure if it was deliberate or not, but I really did think it was a very clever construction, because of the fact that "infinitesimals" really are "infinitely small" values. I read it as a (perhaps unintentional) pun, and it made me laugh, so I commented on it :) No insult intended :)

Stuart.
--
"Forty-two" -- Deep Thought
"Quinze" -- Amélie

[ Parent ]

I didn't think I was insulted (none / 0) (#197)
by nusuth on Mon Jan 10, 2005 at 03:36:56 PM EST

I must have sounded too defensive. Well,among the reason for unintentional typo (actually, mistranlation), grandparent also exlains why sometimes you cannot read my mood from an English comment I made.

[ Parent ]
like I said (none / 0) (#85)
by gdanjo on Thu Jan 06, 2005 at 05:06:12 PM EST

a mathematical solution is accepted - heck, even observation, the thing that science relies on more than any model, proves the paradox non-existent.

The point is that your quote, while technically incorrect (ie: it doesn't "match the mathematics"), is intuitionally valid. Now, if you delve into mathematics you find it's NOT valid - but you can do that without delving into mathematics also, which is the point of the article.

In other words, if you turn the model on itself (by describing the same model with differing initial conditions) you get to intuitionally see why it fails: it's simply not true that "this continues on forever." I think this is easier to see when you put Achilles in front of the tortoise and apply the model, instead of moving laterally into mathematics and then simply saying "no" to the paradox.

If I ask a German mathematician whether the paradox is real and he talks German then tells me "no", that's not a very intuitive solution (for me) now, is it? This is how people not versed in mathematics often feel about mathematical solutions, and this article is just a fancy way of saying this.

Dan ...
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]

Just shut up your idiocy already. (none / 0) (#260)
by coopex on Fri Apr 08, 2005 at 04:58:46 PM EST

If you ask about a mathematical problem, expect a mathematical answer.
If you ask German speaker, expect an answer in German.
Math is cold, logical and precise.  It doesn't care about your intutition or feelings, because they probably don't make something true or not.  Go read Alan Sokal http://www.physics.nyu.edu/faculty/sokal/transgress_v2/transgress_v2_singlefile.html and realize why math and science have such rigourous standards, and why people in those fields have such disdain for those who come in with no knowledge and think they have a better way than whats been worked on by the brightest minds of the past 400 years.

[ Parent ]
Infinite Condition? (none / 0) (#136)
by anthroporraistes on Fri Jan 07, 2005 at 05:59:51 PM EST

I don't know if it is possible to restate the paradox in a way that it both follows and exists. Since it can be shown experimentally that motion is possible in this world, a version that both follows and exists must use a world model that is incompatible with our world, therefore its stated assumptions must be wrong in physical domain, not logical domain. You could write this article with that version but this version isn't it.

But wasn't that the point of the paradox?  To show that there was a fundamental break between logical reality, and our "illusionary" world?  I think he incompatability is intentional, and desired, as a proving point that our reality is flawed and somewhat nonsensicle.  

I think Zeno's school, like Plato later, put emphasis on logic (and idea/rationalism) before physical reality.  It seems a common thread, before Aristotle, that our manifest world is somewhat flawed.  

I don't understand you emphasis on time scale, though.  If for any given finite amount of time, Zeno's paradox would hold true, would that necissarily mean that for any infinite span it would too?  If we pick any moment (M), within the race, and M's state is going to be necisarily consitant within the premise of Zeno's Paradox, then would over infinite iterations of state M be consitent with the conclusion too?  How would the conclusion or state change when one applies the infinite condition?  Don't understand your reasoning, or perhaps I misread.


---
biology is destiny
[ Parent ]

Explanation of relevance of time scale (none / 0) (#164)
by nusuth on Sat Jan 08, 2005 at 10:58:55 AM EST

I don't understand you emphasis on time scale, though.  If for any given finite amount of time, Zeno's paradox would hold true, would that necissarily mean that for any infinite span it would too?  If we pick any moment (M), within the race, and M's state is going to be necisarily consitant within the premise of Zeno's Paradox, then would over infinite iterations of state M be consitent with the conclusion too?

Zeno's reasoning most certainly doesn't hold for any moment in the race. Zeno describes a series of events (Achilles reaching the point tortoise was at the beginning of the interval) while Achilles is behind the tortoise. Zeno goes on to conclude that Achilles never catches the tortoise because there is an infinite number of events before catching up, none of which leads to catching up. That there are infinitly many events happen before catching up, none of which results in Achilles catching the tortoise, is true in a continuous universe. That none results in Achilles catching the tortoise is also tautological, as events are defined to have that property. Zeno's description of events is accurate, that is not where the paradox is.

To illustrate accuracy of Zeno's description of events, consider the case tortoise is as fast as Achilles. You can still define an event as "Achilles reaching the point tortoise was at the beginning of the time interval." There are still infinitely many events that must happen before anything else happens. As before, none of the infinitely many events leads to Achilles catching the tortoise.

The question is: can we exhaust infinitely many events in a finite amount of time? If we can't, same thing happens over and over again, never allowing anything else to happen. That is what happens when Achilles is not faster than the tortoise: a series of non-catching up events must happen before catching up, but the infinite number of catching up events is not exhausted in a finite time, so nothing else ever happens.

If Achilles is not faster, he never catches the tortoise not because the number of events is not finite, but because the time it takes to go through all of them is not finite. "Never" is a statement about time, not cardinality.

OTOH if we can experience infinitely many events in a finite amount of time, something else may happen. If we exhaust one type of events, some other type of event will happen (as no more of the former type of events left by definition.) Zeno never proves that the infinitely many catching up events cannot be exhausted in a finite time interval. He asserts, by virtue of number of events being infinite, they are never exhausted, so nothing else will ever happen. As there is no proof of that point, the paradox does not follow as stated.

[ Parent ]

I refute it thus (2.63 / 11) (#70)
by Zealot on Thu Jan 06, 2005 at 07:47:31 AM EST

(kicks tortoise)

You win (nt) (none / 0) (#119)
by chaz on Fri Jan 07, 2005 at 07:03:55 AM EST



[ Parent ]
Look up Samuel Johnson for more (nt) (none / 0) (#157)
by calumny on Fri Jan 07, 2005 at 11:56:39 PM EST



[ Parent ]
+1 Turtle soup is tasty. (nt) (none / 0) (#73)
by Stavr0 on Thu Jan 06, 2005 at 08:50:46 AM EST


- - -
Pax Americana : Oderint Dum Metuant -- Bis Quadrennia
more than tasty (none / 1) (#148)
by efexis on Fri Jan 07, 2005 at 09:45:00 PM EST

it's convenient too, it comes in it's own bowl!

[ Parent ]
A counterargument without the infinite sum (none / 1) (#84)
by lonelyhobo on Thu Jan 06, 2005 at 04:52:11 PM EST

Space is not continuous, but rather discrete

BAM

yeah, but it's not really discrete (none / 1) (#88)
by LilDebbie on Thu Jan 06, 2005 at 05:49:39 PM EST

it just looks that way because we see using photons.

My name is LilDebbie and I have a garden.
- hugin -

[ Parent ]
loop quantum gravity disagrees with you (none / 0) (#90)
by lonelyhobo on Thu Jan 06, 2005 at 06:02:17 PM EST

Certain aspects of string theory do as well

[ Parent ]
both are physicist day dreams (none / 0) (#94)
by LilDebbie on Thu Jan 06, 2005 at 06:20:18 PM EST

and the quanta is really only defined as the smallest amount of predictable stuff, much like the photon is the smallest observable stuff.

My name is LilDebbie and I have a garden.
- hugin -

[ Parent ]
Do you have any idea what either are? (none / 0) (#98)
by lonelyhobo on Thu Jan 06, 2005 at 06:40:17 PM EST

Anyways, the idea of anything being purely continuous is fairly irrational anyway.

Invoking a theory that has such bizarre effects while calling string theory or loop quantum gravity day dreams is somewhat ironic.

Further, quanta(and Planck's constant) was defined before Heisenberg, and Heisenberg actually followed from the establishment of that lower limit.

[ Parent ]

They are day dreams... (none / 0) (#102)
by Russell Dovey on Thu Jan 06, 2005 at 08:28:45 PM EST

...until they have some good evidence behind them.

When evidence for string theory being correct is discovered I will dance in ugly pants in the comfort of my loungeroom in suburbia.

"Blessed are the cracked, for they let in the light." - Spike Milligan
[ Parent ]

So... (none / 0) (#104)
by lonelyhobo on Thu Jan 06, 2005 at 08:54:01 PM EST

You'll heartily support the calculus behind the infinite sum to be relating to the real world, but the mathematics behind newer physical theories is just too much for you to take?

what

[ Parent ]

Learn to read. (none / 0) (#113)
by Russell Dovey on Fri Jan 07, 2005 at 01:53:54 AM EST

The maths is fine, but until a shred of evidence is produced which relates all that fancy math to reality in an observable way, it's all just intellectual wankery.

Don't get me wrong, it's really interesting wankery. I love string theory and branes and everything, it's  just completely fucking useless until it can be applied to the real world.

"Blessed are the cracked, for they let in the light." - Spike Milligan
[ Parent ]

was that first line a joke? (none / 0) (#130)
by LilDebbie on Fri Jan 07, 2005 at 12:27:09 PM EST

what with a continuous set being made up of irrational numbers and all.

My name is LilDebbie and I have a garden.
- hugin -

[ Parent ]
Out of curiosity... (none / 0) (#107)
by Lisa Dawn on Thu Jan 06, 2005 at 09:42:03 PM EST

how big is a photon anyway?

[ Parent ]
It depends on its wavelength and bandwidth (none / 0) (#125)
by tetsuwan on Fri Jan 07, 2005 at 09:42:05 AM EST

this gives the probability distribution in space. For example, small bandwith photons can be several kilometers long.

Njal's Saga: Just like Romeo & Juliet without the romance
[ Parent ]

Position of a photon (none / 0) (#259)
by coopex on Fri Apr 08, 2005 at 04:47:26 PM EST

I think that you're thinking of the position of a photon, I know that for all physicists know, and experiments show, electrons are point masses, because that's how they behave.  This might mean however, that they're just so incredibly tiny that we can't measure them.

[ Parent ]
that's not a counterargument (2.00 / 2) (#97)
by speek on Thu Jan 06, 2005 at 06:32:39 PM EST

That's a radical modification of one's worldview in order to resolve a "paradox" most people seem to think is idiotic and sophist wankery.

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

No (none / 1) (#99)
by lonelyhobo on Thu Jan 06, 2005 at 06:45:17 PM EST

It's saying that the question is predicated upon the false assumption that space is continuous.  There are theories that support my assertion.

It's like saying that Relativity was just a radical modification of Einstein's worldview that most seem to think is idiotic and sophist wankery.(Can you outrun a beam of light?)

[ Parent ]

yes, there are theories (none / 0) (#122)
by speek on Fri Jan 07, 2005 at 08:15:27 AM EST

And the time of their birth is often referred to as revolutionary.

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

also (none / 1) (#124)
by speek on Fri Jan 07, 2005 at 08:27:23 AM EST

Proving his assumptions false was kind of Zeno's point. Remember, he was trying to point out our everday perception of reality was mere illusion, so he started from people's common assumptions, and tried to demonstrate that they made no sense. So, by altering the initial assumption, you are not invalidating his argument, you are essentially saying "Zeno, you're argument is powerful and valid, but/therefore I start from a different assumption".

The fact that you already had a different assumption can't be helped by Zeno - he lived 2500 years ago, and you have had the advantage of other people inventing quantum physics.

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

Radical? (none / 0) (#109)
by porkchop_d_clown on Thu Jan 06, 2005 at 10:39:08 PM EST

What reason do you have to believe the universe is continous?

Modern physics pretty much states that it isn't - both matter and energy come in the form of discrete packages and current theories indicate that space and time are the same.

Has anybody seen my clue? I know I had one when I came in here...
[ Parent ]

modern physics? (none / 0) (#121)
by speek on Fri Jan 07, 2005 at 08:13:29 AM EST

You do know when Zeno lived, right? And modern physics is still pretty radical compared to an everyday perspective.

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

Yes, I do. (none / 0) (#137)
by porkchop_d_clown on Fri Jan 07, 2005 at 06:14:20 PM EST

Which means I think you insult him by referring to his paradox as wankery.

Has anybody seen my clue? I know I had one when I came in here...
[ Parent ]
reading comprehension? (none / 0) (#141)
by speek on Fri Jan 07, 2005 at 06:44:26 PM EST

You might want to slow down and read the posts people write.

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

I comprehended the original post quite well. (none / 0) (#146)
by porkchop_d_clown on Fri Jan 07, 2005 at 08:07:25 PM EST

You're the one with the attitude problem.

Has anybody seen my clue? I know I had one when I came in here...
[ Parent ]
What current theories? (none / 0) (#258)
by coopex on Fri Apr 08, 2005 at 04:44:14 PM EST

Do you have a source for strong evidence of teh continuousness of the universe, cause I'd really be interested.

Thanks!

[ Parent ]

A simpler debunking (2.71 / 7) (#86)
by boxed on Thu Jan 06, 2005 at 05:12:43 PM EST

Zenon totally ignores time. So what we have is that for each bit of increasingly short distance we have also another thing that shrinks fast: the time to travel that distance. As one shrinks to infinity so does the other. The two simply cancel eachother out. X-X = 0, no matter how big X is.

yes (3.00 / 2) (#96)
by speek on Thu Jan 06, 2005 at 06:30:47 PM EST

That is a much better debunking than saying Calculus explains it.

--
al queda is kicking themsleves for not knowing about the levees
[ Parent ]

The only point I see Zeno making (none / 0) (#95)
by levesque on Thu Jan 06, 2005 at 06:27:08 PM EST

is that there is no reason to think that mathematics has any particular power when it comes to pointing man towards an "Ultimate Truth". The same still applies today, the math and the paradoxes may be of a more learned nature but the general dilemma still stands.

I'm a bit confused about your 1+1=2 explanation, I see it as systematic labelling, a convention, accepted because it helps. The truth is in its accepted accurate representation of perception not in any mathematical truth per say.

The non situations, like the drops of water joining, remind us that we are modelling sensual reality with this stuff, it doesn't seem that we can ever arrive at a perfect model anymore than we can design the perfect eye -what is the perfect scene composed of that this eyes has to be designed for? But I don't mind a "non TRUTH based reality" because determinism is only fine up to a point and there are always questions like "why this rather than that".



wrap around universe? (none / 0) (#111)
by jsnow on Fri Jan 07, 2005 at 12:17:49 AM EST

It is, after all, the opinion of physicists / cosmologists that if you traverse the universe in a straight line, you will eventually end up at the same spot that you started.

Are there any good arguments for or against a finite universe? I haven't heard any, but it may just be my ignorance.

On a side note, any article that expects to stand up to intellectual scrutiny ought to refrain from appealing to "commonly held opinions" when those opinions are in fact controversial or not well known. I would accept the statement "It is, after all, the opinion of geographers / astronomers that the earth is round", but anything even slightly less universally accepted ought to be backed up by citations. (Other phrases to avoid include "It is self-evident that", "It is well known that", "Experts believe that", etc...)

Well...general relativity (none / 0) (#120)
by GreyGhost on Fri Jan 07, 2005 at 07:57:47 AM EST

Assumes implicitly that the universe is 4-dimensional and not embedded in a higher dimension. What the geometry of the space is like however (flat, negatively-curved or positively curved) depends on the mass in the universe (and all that unaccounted dark matter and stuff). Assuming you believe GR.

If it's negatively-curved - then the universe will keep expanding forever (into what - who knows....because GR doesn't give us the tools to say what's outside of the universe and it might not even make sense to even ask the question) but I would say in that case the statement is wrong. To see why - try doing something like playing chess on a hyperbolic plane (like those MC Escher paintings with the birds where the birds get smaller and smaller the farther you move towards the edge). A hyperbolic plane is a negatively curved space. Your knights are going to do things like spiral off to infinity and they will never be able to come back to their starting point. If you don't move them very far....then maybe you can have your knight return to where it started at...but if you move your knight too far away you are screwed.



[ Parent ]

Universe defined (none / 0) (#156)
by efexis on Fri Jan 07, 2005 at 11:13:25 PM EST

"then the universe will keep expanding forever (into what - who knows....because GR doesn't give us the tools to say what's outside of the universe and it might not even make sense to even ask the question"

Does defining the word "universe" help? Lets say, the universe is the sum of everything that can be defined as not-nothing (mass, energy, everything we know of, and everything we don't). Therefore, what's outside the universe, is nothing. It must be. If it wasn't nothing, then it would be something, thus included within our definition of the word "universe", so could not be outside the universe, no? If you flew to the edge of the universe, and then flew a bit further, you would become the edge of the universe. There would be nothing between you and what was the edge, and there'ld be the same nothing in the outwards direction. There'd be no limit to how far you can fly, as it would take "something" to stop you.

SO... for there to be anything outside the universe... we must redefine (or correct me on my definition of) the word Universe, otherwise accept that there must be nothing outside the universe. (ouch)

-2A

[ Parent ]
Nothing is something.... (none / 0) (#178)
by GreyGhost on Sun Jan 09, 2005 at 12:07:27 AM EST

And right now we don't know enough about the nothing we can study to form conjectures on the nothingness outside of the nothing we can't study.

But that's kind of beside the point. Einstein was just applying Occam's Razor when he formulated his general theory. Basically a theory should be as complex as needed to describe physical phenomena, and not any more complex. Since we can't visit the edge of the universe...and whatever goes on outside the universe we are in doesn't seem to influence behavior inside the universe we are in in a way we can yet detect....don't worry about it and don't start putting 5-dimensional spaces into your gravitational theories.

The same principle guided scientists when they first discovered planets and assumed them to be as smooth and perfect as billiard balls. Most scientists probably felt that this probably wasn't the case....but there was no way to refine the model until better telescopes were invented so why deal with other planets having volcanos and mountain ranges until the equipment was there to actually study these things?



[ Parent ]

They're mutually exclusive! (none / 0) (#180)
by efexis on Sun Jan 09, 2005 at 04:37:46 AM EST

You're misusing the word "nothing" to mean "nothing known or relevent", ie, if there is something outside our universe and it's not affecting us in any way, it shouldn't/couldn't be included in our theories/formulas/etc, which I would agree with. If on the other hand it does affect us, we can realise something's out there, just as we realised pluto was there, without actually having to see it.

So while we could prove there's something out there, we could never prove that there's nothing (as opposed to nothing relevent) out there. But there IS still a difference, nothing is not something! :-)

- 2A

[ Parent ]
agreed (none / 0) (#155)
by gdanjo on Fri Jan 07, 2005 at 11:04:21 PM EST

the "intuitive solution" - where the "negative" of the fable is as much a paradox as the original - relies heavily on that quote; in fact, I had written in "physicists / cosmologists" because I didn't know which was more applicable, and I had planned on backing up the assertion with some citations. I just plain ran out of time / patience, and so left the statement at that.

In my defense, all I can say is that the story panders to the intuition using intuitional methods, and therefore does not necessarily require mathematical rigour. I can't say this answer makes me comfortable, but hey, this is "only" k5. :-)

Thanks for your input.

Dan ...
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]

I was going to post a similar quibble (none / 0) (#191)
by JetJaguar on Mon Jan 10, 2005 at 01:51:32 AM EST

The idea that space turns in on itself so that if you travel far enough that you will wind up back where you started is really more of a plausible conjecture. No physicists or cosmologists really hold a positive or negative opinion on the idea, it's just one of several plausible possibilities.

As an astronomer myself, I certainly wouldn't say that I believed that this is in fact how the universe is shaped. I would say that it's one possibility that may be true, but we really don't know enough about the "real" geometry of the universe to say for sure, although I think the more recent data indicating that the universe is open does probably put this idea to rest.

At any rate, it's a minor quibble, and I don't think that this really makes any difference in the context in which you use it. I think the main point is that if there are two objects seperated by a very large distance, does it really make sense to any longer say that one object is in front of the other? If I'm driving east in San Diego, and there's someone else driving east in Atlanta, do I gain anything useful from knowing that the guy in Atlanta is "ahead" of me? Does the statement that the guy in Atlanta is ahead of me even make sense in this context? The distances are so far compared to the speeds involved that it really doesn't matter anymore, which I think is the point you're trying to make (more or less).

[ Parent ]

"Once and for all?" (2.00 / 3) (#127)
by OmniCognate on Fri Jan 07, 2005 at 11:26:58 AM EST

Infinitesimals were only banished from rigorous mathematics until the sixties. Their use has been put on a rigorous foundation with nonstandard analysis.

In the meantime, infinetesimals were frequently used in informal arguments in calculus. The rules prescribed by nonstandard analysis for using infinitessimals safely are essentially the same "commonsense" rules that are used in the informal arguments, together with a few additional subtleties necessary to guarantee rigour.

Standard analysis, using no infinitesimals, is highly unintuitive, but relatively (repeat, relatively) easy to provide a rigorous framework for. Nonstandard analysis, in contrast, produces a set of rules that match quite closely to our intuitive feelings about infinitesimals and infinities, and which are consistent. However it is a tricky job to derive those rules in a rigorous fashion from accepted axioms.

In that sense, then, you could perhaps argue that the only reason mathematicians had to temporarily abandon infinities is because their intuitive notions, far from being naive, were too subtle to be easily captured in a rigid mathematical framework.

Such a framework is required in order to avoid nonsensical results and to resolve confusions such as Achilles and the tortoise, but in the free-flow of actual thought, before analysis (of either type), it was intuitive reasoning with infinitesimals that led to the development of most of calculus. The application of rigour came later.

Intuition is a deeply weird thing.



nonstandard analysis requires a different logic. (1.00 / 5) (#142)
by the ghost of rmg on Fri Jan 07, 2005 at 06:54:35 PM EST

in it, one loses the law of the excluded middle, which means no proof by contradiction. that is not a subtlety. there is nothing intuitive about nonstandard analysis unless you learned mathematics from a physicist -- that is to say, not at all.


rmg: comments better than yours.
[ Parent ]
How does ex. middle break proof by contradiction? (none / 0) (#221)
by sab39 on Fri Jan 14, 2005 at 03:58:05 PM EST

How is [X OR NOT X] required to be true for proof by contradiction?

Proof by contradiction simply requires that [X AND NOT X] be false, which is a different statement. Perhaps you lose that too, but I don't see how it's a consequence of losing excluded middle.
--
"Forty-two" -- Deep Thought
"Quinze" -- Amélie

[ Parent ]

It doesn't (1.50 / 2) (#226)
by OmniCognate on Sat Jan 15, 2005 at 10:56:27 AM EST

The grandparent is a troll, as evidenced by the fact that he modded me 0 after a while when he realised I wasn't going to reply to him.

There is a logic restriction in nonstandard analysis, but it applies not to the types of proof you can use, but to which results can be brought over from standard analysis. Roughly speaking it says that any statement in first order logic that is true for the real numbers is true for the hyperreal numbers as well.

What this means is that not everything you can say about the real numbers holds true when you include infinite and infinitesimal numbers, which is pretty obvious when you think about it. The restriction (which is part of what is called the transfer principle) provides a way of telling which statements carry over and which don't.

Within nonstandard analysis itself, proof is conducted in exactly the same as in any other branch of mathematics. There most certainly is no "different logic". The law of excluded middle applies as always and, as you pointed out, even if it didn't there would be no problem with proof by contradiction.

As for intuitiveness, I never claimed the full rigorous study of nonstandard analysis was intuitive. I actually pointed out how difficult it was. What I claimed was intuitive was reasoning with infinitesimals. I also claimed that nonstandard analysis provides (ie. can be used to prove the validity of) a set of rules which can be used when reasoning with infinitesimals. These rules match closely with the intuitively "obvious" properties of infinitesimals (for example any real number divided by an infinite number yields an infinitesimal), but with a few extra subtleties added in order to avoid nonsense results. One of these subtleties is the transfer principle. It is a subtlety, and it really isn't hard to get your head around in order to use in practice. Of course, it is very hard to prove, but that's already been done by Abraham Robinson :-)

For an introductory calculus course based on the rules furnished by nonstandard analysis, see Keisler's book "Elementary Calculus: an Infinitesimal Approach", which is out of print but available free online here.

[ Parent ]

for the record, (none / 1) (#227)
by the ghost of rmg on Mon Jan 17, 2005 at 01:45:57 AM EST

i zeroed you because you are an imbecile who clearly does not know what he is talking about. any elementary text on nonstandard analysis will reveal that it is as i said. that is why it has not been widely adopted in the mathematical community.

didn't you ever wonder why some results from the rest of mathematics are incompatible with nonstandard analysis?


rmg: comments better than yours.
[ Parent ]

IHBT (none / 1) (#232)
by OmniCognate on Mon Jan 17, 2005 at 09:19:57 AM EST

Here is the source of what I know. I have read the brief but excellent introduction to nonstandard analysis here and, interested by what I saw, I tracked down and read a physical copy of the elementary calculus textbook to which I linked in the post above. This certainly doesn't qualify me as having more than the most rudimentary understanding of the subject, but I contend that I know a damned sight more than you.

If you disagree, I would like you to post a link to a freely available online text or paper that supports your claim (a result as fundamental as the law of excluded middle not applying in nonstandard analysis should be easy to find free online - try eg. mathworld).

For the record, the situation is not that "some results from the rest of mathematics are incompatible with nonstandard analysis". The situation is that some results that are true for the real numbers are not true when infinitesimals and infinities are included.

This is similar to saying that not everything that is true for the integers is true for the reals (for example, the statement "there exist two numbers x and y, y > x such that no z exists for which y > z > x," is true for the integers but not for the reals). We do not say, as a result of this, that "some of the results from number theory are incompatible with real analysis". We most certainly do not conclude that real analysis "requires a different logic" to number theory, or that either number theory or real analysis "have not been widely adopted in the mathematical community" as a result.

If you think I'm wrong, then link to someone who supports your viewpoint. Calling me an imbecile does not make you any more right, and the fact I eventually bit does not make you any less of a troll.



[ Parent ]
do your fucking homework. (none / 1) (#235)
by the ghost of rmg on Mon Jan 17, 2005 at 04:58:17 PM EST

a google search for "nonstandard analysis" "excluded middle" tells you all you need to know. once again, "for the record," you are arguing with someone who knows a "damn sight" more about logic and mathematics than your sorry ass.

i apologize for being galled by some debutante who read an article on nonstandard analysis in readers' digest and decided he was an expert on the subject. i do not apologize for saving the readers here the trouble of learning nonstandard analysis only to find after several hours wasted what you should already know: that the law of the excluded middle, and therefore proof by contradiction is lost. i.e. that perhaps the most powerful tool of classical logic must be thrown away just because you're too lazy to do real analysis.


rmg: comments better than yours.
[ Parent ]

No link, you lose [n/t] (none / 0) (#237)
by OmniCognate on Tue Jan 18, 2005 at 05:01:41 AM EST



[ Parent ]
Ah, I think we've both been confused (none / 1) (#238)
by OmniCognate on Tue Jan 18, 2005 at 05:27:04 AM EST

It turns out there is another form of infinitesimal analysis called Smooth Infinitesimal Analysis. This is distinct from nonstandard analysis. Smooth infinitesimal analysis is indeed incompatible with the law of excluded middle. The two are both mentioned (as separate theories) in the wikidpedia article about infinitesimals.

I'll offer an olive branch here. When you replied to my original article, what you said sounded so utterly, unfeasibly wrong to me that I figured you either had to be a nutter or a troll. I gave you the benefit of the doubt and didn't reply. When you undeservedly modded me zero, I figured you had to be a troll and were just pissed off that I hadn't replied. It seemed a pretty spiteful thing to do.

Now it seems that while you were wrong (nonstandard analyis does not require any weird logic), you were close enough to having a valid point that I may have been unfair to accuse you of trolling.

So, I apologise for calling you a troll. However, once again, while there is a form of infinitesimal analysis that is incompatible with the law of excluded middle, nonstandard analysis isn't.



[ Parent ]
ugh... (none / 1) (#228)
by the ghost of rmg on Mon Jan 17, 2005 at 03:02:02 AM EST

you lose the equivalence between (A v ~A) and (A => A), which breaks the law of contrapositives.


rmg: comments better than yours.
[ Parent ]
Wrong on so many levels... (3.00 / 6) (#132)
by Infophreak on Fri Jan 07, 2005 at 02:10:54 PM EST

Allow me to write an attempt at a translation of this article (which is in Danish) from FAMØS which is the magazine of the institute of mathematics at the university of Copenhagen.

I think it is a much better explanation of the paradox (and at the same time a nice hit in the head to humanists who think they know a thing or two about math):

"When Humanists do Math

When I recently (in my capacity as a journalist for FAMØS) was on a trip to the so-called real world, I used the oppurtunity to flip through a pop-philosophical book named "Politikens bog om de store filosoffer (Politikens forlag, 1999)" (i).
In this book the greek philosopher Zeno is mentioned - a name any mathmatician invariably will connect with Zeno's paradoxes. The book goes:

"One of these paradoxes is the story of Achilles and the tortoise [...] (ii)"

So far we have an objective and reasonable recount of how Zeno's arugment was, and thus perfectly sound as a piece of history of philosophy. But then the tragedy begins. As I read the following, I grew so tired that one should hardly think that I had not slept since the days of Zeno:

"The point is that we are facing a flawless logical argument that none the less leads to a false conclusion. [...] (iii) Some persons can actually be quite disturbed by this. Something must be wrong with the logos, they say. But noone has yet been able to put the finger on where the problem is."

To further wave the red rag they end with the following

"Perhaps it will be solved some day just like we finally now, after ~400 years of speculations, have the solution to Fermat's last theorem."

What truly made me sad was really not that some incompetent humanist tried to tell me that he knew something about mathematics while demonstrating the opposite. It was more that the last approximately 2500 years of matematical development that seperates Zeno from us seemed to have gone completely unnoticed to the world outside the matematical community.
I will therefore dedicate the rest of this article to handing my reader weapons with which to go out and inform the (apparently) unknowing masses on what actucally happened when Achilles and the tortoise raced. We will have to save Fermat's last theorem for another time.

Let us see what we are dealing with: Achilles and the tortoise run both with constant speeds and their positions relative to some point of origin can therefore be described by to growing linear functions, A and T (iv). Their dependance of time can be described by these expressions:

A(t) = vt + A(0),
T(t) = wt + T(0),
a,s > 0, t >= 0,

where t is time and v and w respectively are Achilles' and the tortoise's velocities.
Since the placement of zero on the axis has no significance (it will only shift the entire problem with some constant), we can place it such that A(0) = 0. We can also use Achilles' velocity as unit such that v = 1 and as we knew that the tortoise ran half as fast as Achilles, w = 1/2.
Personally I think it's rather unimpressive of this great greek hero that he cannot run more than twice as fast as a mere tortoise, so let us instead take w = r where 0 < r < 1. Then the reader can make the race as fair or unfair as he or she would want.
The two combattants' placements are now given by the expressions:

A(t) = t,
T(t) = rt + T(0),
0 < r < 1, t >= 0.

By setting t = 0 we see that the tortoise's head start must have been T(0) which consequently must be greater than 0. (There most be a limit to the unfairness!)

Some would now solve simply solve the equation T(t) = A(t) graphically or symbolically with respect to t and conclude that Achilles overtakes the tortoise when t = T(0)/(1-r). I think that's a slightly too easy way to avoid the problem, and I will therefore attack it from another angle which I think better explains the paradox.

Let us try to relate to way Zeno does things: When Achilles has travelled the distance of the head start, T(0) (which he has done at the time t = T(0)), the tortoise has pulled further rT(0) ahead. Achilles travels this distance in rT(0) and reaches this new point hwen t = (1+r)T(0). In this period the tortoise moves further r2T(0) ahead. When Achilles has reached this point t = (1+r+r2)T(0) and so on.
The general system begins to appear: Zeno constructs a sequence of points in time t0, t1, t2,... given by the expression

tn = ($sum_{i=0}^n r^i) T(0)$ (v).

And then he observes that every time n is increased by 1, the distance T(Tn - A(tn) is multiplied by r (in the quotation it was halved), but it never becomes 0. Zeno's argument can now be restated:

There is no natural number n such that T(tn) - A(tn) = 0,
hence there is no real number t such that T(t) - A(t) = 0.

When it is stated like this it suddenly becomes clear that Zeno concludes more than he has argued for. He has certainly pointed out infinitely many points in time where the tortoise is ahead of Achilles, but that's no argument for concluding that it never occurs.

[...] (vi)
"

(i): "Politikens forlag" is the publisher, and the title means "Politiken's book on the great philosophers".

(ii): I will not translate this part. It's just a statement of Zeno's argument.

(iii): His edit, not mine.

(iv): "S" in the original for "skildpadde", the Danish word for tortoise.

(v): Sorry for my use of LaTeX, but there is no sane way to write a sum in HTML. You can see the symbol in the original article, of course.

(vi): The rest is just some simple computations that are irrelevant for this post.

PS: OMG that ended up being really long, and I don't feel like checking it for typos and/or moronic errors. But it should be mostly OK

*smack* (none / 0) (#144)
by Infophreak on Fri Jan 07, 2005 at 07:29:02 PM EST

That's the sound of me hitting myself for forgetting to mention that the article is on page 19 in the PDF file. I don't suspect that many has read it - this being a site in English and all, but I always feel stupid when I forget those small details. I wish there was a way to make a link to a page of a PDF file (is there?)

[ Parent ]
Don't worry about it. (3.00 / 5) (#147)
by SIGNOR SPAGHETTI on Fri Jan 07, 2005 at 08:54:38 PM EST

The article is wrong and beside the point. Your mathematician friend forgot to mention (1) what metaphysics fairy told him it makes sense to speak of an instant t in a system of two moving bodies, and (2) what math fairy told him his function is constant at t. Maybe the people in Norway believe very strongly their calculus teacher's intuition that objects in motion have precisely defined positions at each given "instant" in time, but that don't make it physics.

The summation of an infinite series to explain Zeno's paradox is a useful pedagogical device that by coincidence gives the correct numerical answer, BUT IT DOES NOT RESOLVE THE PARADOX OR EXPLAIN HOW MOTION IS POSSIBLE. All it demonstrates is that mathematicians are notoriously stupid about physics and what math means.

Zeno's paradox is "resolved," to the extent a metaphysical artifice that competes with our intuition can be said to resolve anything, by the assumptions of Lorentzian invariance and the relativity of space and time. Finally, read Wittgenstein and Mach, learn that this shit don't matter.

--
Stop dreaming and finish your spaghetti.
[ Parent ]

*sniff* (none / 0) (#234)
by warrax on Mon Jan 17, 2005 at 01:57:16 PM EST

That was... beatiful.

-- "Guns don't kill people. I kill people."
[ Parent ]
in the domain of mathematics (none / 0) (#154)
by gdanjo on Fri Jan 07, 2005 at 10:46:24 PM EST

the problem is solved. I say as much in the article.

The problem, however, comes from a) the translation of the fable into mathematics, and b) the interpretation of the mathematical conclusion back into a natural-language conclusion.

The mathematics says "there's no paradox", but the mathematician offers no interpretation as to what this means to the fable - what's wrong with our method of evaluation?

To be honest, I find the confidence and arrogance of mathematicians such as the one you quote on par with the requirement in the past that you know latin to know anything. People thought latin was the universal language of knowledge, but it turns out all languages are as good as each other. Similarly, mathematicians believe mathematics is the universal language of truth, but forget that mathematics is only as good as it's interpretation.

Bottom line is this: Zeno's paradox is interesting outside of mathematics, but to the mathematician there is NOTHING outside of mathematics, and so he attacks all these opinions with mathematical tools. Rinse, repeat, then add to the list of "grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms" (Russel).

Dan ..
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]

you didn't get A's in math, right ? (none / 0) (#160)
by fhotg on Sat Jan 08, 2005 at 02:51:45 AM EST

Your attack on a mathematical point of view is besides the point. While you are right that above article doesn't "solve" Zenos paradox, only someone with some abilities in mathematical logic can see and show that there is more behind the paradox than Zenos ignorance of infinite series.

Bottom line is this: Zeno's paradox is interesting outside of mathematics, but to the mathematician there is NOTHING outside of mathematics, and so he attacks all these opinions with mathematical tools.

Opinions cannot be attacked with mathematical tools.

From famous mathematicians that come to mind it is much harder to pick one who definitely did not contribute to philosophy than the other way round.
~~~
Gitarren für die Mädchen -- Champagner für die Jungs

[ Parent ]

Context jumping: a basic mistake (3.00 / 9) (#133)
by efexis on Fri Jan 07, 2005 at 03:26:22 PM EST

"Surely, after travelling approximately the distance of the earth's circumference Achilles will no longer be in-front-of the tortoise, but behind!"

This is a basic mistake, you cannot escape the context you're describing to make a point within the context. Physical location has nothing to do with your position within the context of the race. If you nearly lap someone on the track of a long distance race, you may be physically behind them (defined by being able to see their back, rather than they seeing yours), but you are still ahead in the race. Otherwise, being 200m ahead of someone on a 400m track, would be defined as a draw (both equally ahead and behind)!

You're using rules outside the context that you're trying to describe. The fact that the world is round makes no difference to the fact that the winner of a race is who completes a distance in less time than the other. Just as two drops of water merging when they touch is not within the context of the description 1+1=2. One drop of water, plus one drop of water, DOES equal two drops of water. Your argument of drops merging has as much weight as saying one apple, plus one apple, doesn't equal two apples, as you might get hungry while bringing them together and eat some. It's not included in the equasion, therefore should not be considered as part of the "whatever" you're describing.

But thanks for making me think :-) Hope this does you too.

agreed (none / 1) (#153)
by gdanjo on Fri Jan 07, 2005 at 10:32:24 PM EST

I totally agree with your post. The problem is that context jumping occurs for every single statement including the translation of a fable into mathematics. My point is that a mathematical solution is not intuitive because of these "context jumps."

One drop of water, plus one drop of water, DOES equal two drops of water.
You are now making the exact same "context jump" that you see in me: in the physical world, we consider objects "countable" (ie: representable by numbers) via measurements. In the intuition, a "drop" of water is a structural measurement, NOT a volumetric measurement. Therefore, the merging of two structural drops of water results in one structural drop of water - this is true at the same time as the volumetric merging.

Similarly, the statement that "Achilles never overtakes the tortoise" is a structural statement in that it is saying "Achilles will never be ahead of the tortoise" which is wrong because near the "finish line" one cannot use the same logic and still say the tortoise is ahead - they will come up against a fundemental limit to the tortoises forward movements that the mathematics of limits describes.

Your argument of drops merging has as much weight as saying one apple, plus one apple, doesn't equal two apples, as you might get hungry while bringing them together and eat some. It's not included in the equasion, therefore should not be considered as part of the "whatever" you're describing.
The thing is, it is included in the intuition (which is just a predictative reflection of reality), and therefore needs to be accounted for. In theory you can say with ABSOLUTE CERTAINTY that if I give you 1 + 1 apples, you will recieve 2 apples. In PRACTICE, the earth could be swallowed up by the sun, and you never get your apples.

Now, for this extreme example, we can ignore the probability of the sun swallowing us. Similarly, for the most part we can ignore weirdness when travelling so fast that you approach the speed of light - but at some point, for some context, they need to be considered (in my sun-swallowing example, you need to redefine what "absolute certainty" means)

But thanks for making me think :-) Hope this does you too.
Indeed it did. I just wish I had this kind of feedback before writing the story so I could target it better. It's really difficult to attempt to guess how people will take a story before it's written, and then target it at this imaginary audience.

Your comments are much appreciated.

Dan ...
"Death - oh! fair and `guiling copesmate Death!
Be not a malais'd beggar; claim this bloody jester!"
-ToT
[ Parent ]

Not agreed (none / 1) (#158)
by efexis on Sat Jan 08, 2005 at 12:20:21 AM EST

Thanks for the reply (this is in fact my first K5 debate ;-)) but I feel you're still missing the point. Maths is an absolute description language. It's here we hit the problem - relying on our senses, we appear to not be able to be absolutely sure of anything. But, maths is absolute, so it must get round this. How? By defining everything that it describes (which means that ultimately, it says nothing!)

Let's talk about one plus one. It would be such a pain to say "one plus one" everytime we wish to talk about it, so we define a shortcut, an abbreviation; the word "two". Follow:
  • "Two" is defined as "one plus one", so
  • "One plus one" has the same definition as "two".
    (shorten "has the same definition as" to "equals"):
  • "One plus one" equals "two".

    So you see, the only way we can know that one plus one equals two, is because we have defined them both to mean the same thing. One drop of water, plus one drop of water, MUST equal two drops of water, because two is defined as one plus one. If you put the two drops together so they merge and forme one drop, it could no longer be called "two drops", therefore could no longer be called "one drop plus one drop", therefore describing it using those words would be false.

    Make sense?

    (It's also worth noting that the statement "one drop of water, plus one drop of water, equals two drops of water" would REMAIN true, even if there were no such things as water or drops)

    - 2A

    [ Parent ]
  • yeah (none / 1) (#163)
    by gdanjo on Sat Jan 08, 2005 at 06:33:00 AM EST

    If you put the two drops together so they merge and forme one drop, it could no longer be called "two drops", therefore could no longer be called "one drop plus one drop", therefore describing it using those words would be false.
    Once again the problem here is in the definition. Let's try the two drops of water experiment again: let's say I "add" one drop of water to another (which translates in reality to placing two separate drops of water within the same frame of reference). I have now added 1 + 1 and I have two. Now let's say that I merge these two drops of water back into one drop - have performed a minus operation? Mathematically, it seems like I have, because (as you admit) I now have 1 drop of water. Intuitively, I really havn't.

    So, either I performed a plus operation as well as a minus operation, or I did nothing at all. In both cases, we have dissonance between mathematics and reality, as well as a dissonance between mathematics and our intuition. While we may never be able to resolve such quandries, it doesn't usually matter because, in the end, the shortcuts that mathematics takes do not have an effect on the outcome, and so we can usually ignore these inconsistencies. But that doesn't mean they don't exist.

    It's important to remember that mathematics is a subset of reality - a subset that happens to be exteremely useful, and when contorted in the right way, seems to be able to describe all of reality. But it's this contortion that makes it un-intuitive, which is why we don't speak mathematics in everyday conversation.

    I understand what you're getting at, and I do appreciate that most mathematicians don't consider "1+1=2" to be absolute truth but just a shorthand way of saying "this operation is true because it is SO GODDAMN useful, we'd be stupid to treat as not very special."

    Still, that's no reason to throw out non-mathematical (eg: intuitive) descriptions, and that's no reason that we should simply ignore our intuition and only "talk maths" - which seems to be the conclusion we must reach if we only accept the mathematical solution to the paradox.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    maths is definition (none / 0) (#167)
    by efexis on Sat Jan 08, 2005 at 03:11:49 PM EST

    Now let's say that I merge these two drops of water back into one drop - have performed a minus operation?
    No you haven't. The resulting drop would not be equal (in all ways) to either of the original drops. You could use maths to describe the volume of water: 1u+1u=2u (where u is the unit of measurement). This statement would remain true whether you merge the drops together, keep them seperate, whatever, as you're describing how much water is there. To describe the drops mathmatically, you could define:
    a = '1u drop of water'
    b = '2u drop of water'
    so:
    1a + 1a = 2a or
    1a + 1a => 1b, but NOT
    1a + 1a = 1b, because '=' means 'is identical to'. If you do say that 1a + 1a = 2b, then you must also say that b=2a, and that a='1u of water', not a='1u drop of water'.
    So, either I performed a plus operation as well as a minus operation, or I did nothing at all
    No, you have performed an action, that changes what you have defined, into something that no longer fits that definition. It cannot be one drop of water, AND two smaller drops of water, at the same time. Unlike programming languages (or any type of instruction language), the equals sign doesn't say "give the left the same definition as the right". It says they already have the same definition. If they don't, then the statement, simply isn't maths.
    It's important to remember that mathematics is a subset of reality
    No, it's important to forget that, as it's untrue. Maths is a subset of logic, it's a language of circular definition. Equations just say that one way of saying something, is the same as another way of saying the same thing. It doesn't say whether something exists, behaves in a certain way, or relates to "reality" in any way at all.

    - 2A

    [ Parent ]
    hmm (none / 0) (#172)
    by gdanjo on Sat Jan 08, 2005 at 09:18:02 PM EST

    I think we're getting way off track here. You say:

    If they don't, then the statement, simply isn't maths.
    Exactly. If you re-read the story, you'll see that I translate the mathematical statement "1+1=2" into the natural language statement "Adding one thing to another yields two things." From this I conclude that in natural language the statement "1 drop of water + 1 drop of water" equals "1 [larger] drop of water", which is a valid answer in the natural language domain.

    It seems that you're trying to convince me of the mathematics of it, but really I don't need convincing. My point is that translation into natural language statements introduces ambiguity that plays with our intuition, which is why Zeno's paradox still seems paradoxical, but really isn't.

    Your original reply also questioned the race and my interpretation of it. I talk about this in my diary - again, the confusion comes about because my point is that translation of mathematical statements into natural language introduces ambiguities inherent in natural language itself; they are NOT questioning the validity of the mathematics, they are merely questioning the layperson's interpretation of it.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    Nearly there! (none / 0) (#179)
    by efexis on Sun Jan 09, 2005 at 03:49:27 AM EST

    you'll see that I translate the mathematical statement "1+1=2" into the natural language statement "Adding one thing to another yields two things."
    This is not ambiguity, this is plain error. Your "translation" describes an action; a process; cause and effect. You introduce a second state/frame of reference; what you have before you perform the action, and what you have after.

    Consider a more correct translation: "one thing and another thing is two things". Note the lack of a verb. Your second statement would therefore become "one drop of water and another drop of water is two drops of water". No verb, no possible interpreted transformation, no alternate frame of reference, no ambiguity, no problem.

    Whilst at this level the error appears very slight, technically everything you've base upon your "translation" becomes fundamentally flawed. You cannot blame language ("ambiguities inherent in natural language itself"), OR question interpretions of the translation ("questioning the layperson's interpretation of it"), for an error you've added to make a point!

    Yes this is splitting hairs... but isn't that what all this is about? :-p

    Wow I need to go to bed!

    - 2A

    [ Parent ]
    nope, missed the bus completely (none / 0) (#185)
    by gdanjo on Sun Jan 09, 2005 at 06:03:05 PM EST

    you'll see that I translate the mathematical statement "1+1=2" into the natural language statement "Adding one thing to another yields two things."

    This is not ambiguity, this is plain error.

    Which brings us right back around to the problem in the first place. If you ask anybody on the street whether "1+1=2" is the same as "Adding one thing to another yields two things", 99% of them will say 'yes, they're basically the same thing.' Only mathematicians (and other -icians) will have the rigorous evaluation discipline necessary to recognise the subtelties introduced in this translation. And I make this point clearly in the story.

    Any methodology which calls the above translation a "plain error" is most certainly not intuitive. And this is the whole point of the story.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    Leading question (none / 0) (#187)
    by efexis on Sun Jan 09, 2005 at 08:26:11 PM EST

    If you ask anybody on the street whether "1+1=2" is the same as "Adding one thing to another yields two things", 99% of them will say 'yes, they're basically the same thing.'
    This is a leading question, it's still you putting the verb in the translation to try and make a point. If you ask the person to translate on their own, they're more likely to say "one add one is two", "one thing plus one thing is two things" or so on. I'd bet that most people would only include the verb is you asked them something like, "how would you calculate this?"

    You can ask joe public leading questions to prove anything.

    You've been trying to say that translation causes ambiguity, but that's plain not true, it's the translator who has deliberately mistranslated in an attempt to mislead the reciever, to create the illusion of an ambiguity or paradox that isn't really there. The problem may be where you say it is, but it's not what you say it is.

    [ Parent ]
    ugh (none / 0) (#188)
    by gdanjo on Sun Jan 09, 2005 at 09:01:43 PM EST

    You've been trying to say that translation causes ambiguity, but that's plain not true, it's the translator who has deliberately mistranslated in an attempt to mislead the reciever, to create the illusion of an ambiguity or paradox that isn't really there.
    If that's what you think, then that's what Zeno did. He deliberately told the fable in such a way that he would cause the translator (ie: the one who reads the fable) to mistranslate and imagine a paradox.

    Whatever is the cause of it, these translational errors occur - should, would, could don't enter into it. The intuition does.

    The problem may be where you say it is, but it's not what you say it is.
    Time and again you come back and say "no yuo! are mathematically incorrect" and time and again I say "I know, but I'm talking about intuition."

    Now, if you do not refute the following:

    Any methodology which calls the above translation a "plain error" is most certainly not intuitive. And this is the whole point of the story.

    then there is nothing more to discuss.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    I do! (none / 0) (#189)
    by efexis on Mon Jan 10, 2005 at 12:00:52 AM EST

    He deliberately told the fable in such a way that he would cause the translator (ie: the one who reads the fable) to mistranslate and imagine a paradox.
    If the paradox exists in Zeno's description of the problem, but not in reality, then it is his translation of reality that is inaccurate. If his description of reality is inaccurate, then he is the faulty translator, not the reader. If his description of reality is accurate... then the paradox is real, and Achilles will never overtake the tortoise.
    Now, if you do not refute the following:
    "Any methodology which calls the above translation a "plain error" is most certainly not intuitive."
    then there is nothing more to discuss
    I immediately and instinctively knew that your translation was false, and it took some thought before I realised the nature of the problem in your added verb. I certainly wasn't looking for a problem when I read the artical, I was in fact quite enjoying it! So, what is intuitive?

    Intuitive: 1. Of, relating to, or arising from intuition.
    2. Known or perceived through intuition. See Synonyms at instinctive.

    Intuition: n, 1a. The act or faculty of knowing or sensing without the use of rational processes; immediate cognition.

    So, occording to the dictionary, it was intuitive, so I guess I do refute... I can't argue with the dictionary.

    - 2A

    [ Parent ]
    oh man (none / 0) (#198)
    by gdanjo on Mon Jan 10, 2005 at 07:23:50 PM EST

    If the paradox exists in Zeno's description of the problem, but not in reality, then it is his translation of reality that is inaccurate.
    Although I have not raced a tortoise specifically, I'm quite confident the paradox does not exist in reality.

    I immediately and instinctively knew that your translation was false
    This is quite a natural response, given that most of us know that the paradox does not exist in reality.

    and it took some thought before I realised the nature of the problem in your added verb.
    Given that it "took some thought" to realise what the error was (note that this is different to instictually realising that there is an error), and given your definitions of intuition, I would conclude that the error itself is neither instinctual nor intuitive. The existence of the problem is most definately intuitive - THAT'S THE WHOLE POINT OF THE FABLE - but the solution is not.

    I'm not sure what else I can add, or whether I can say this same thing any more times than I already have.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    oh boy (none / 0) (#201)
    by efexis on Tue Jan 11, 2005 at 07:15:48 AM EST

    If the paradox exists in Zeno's description of the problem, but not in reality, then it is his translation of reality that is inaccurate.

    Although I have not raced a tortoise specifically, I'm quite confident the paradox does not exist in reality.
    Then you agree his translation of reality must be inaccurate... which, as what I said, him the the faulty translator.
    I immediately and instinctively knew that your translation was false

    This is quite a natural response, given that most of us know that the paradox does not exist in reality.
    Who's talking about the paradox here? Your translation was of your example, involving 1+1=2. It was this I called "a plain error". You said me calling it "plain error" was unintuitive, to which, I replied, it was not, as it was my intuition that pointed out to me the error was there. Further cognition was required to word it, to tell you why it was an error.
    or whether I can say this same thing any more times than I already have
    I suggest that you don't!

    [ Parent ]
    Thanks guys! (none / 0) (#202)
    by generaltao on Tue Jan 11, 2005 at 12:05:32 PM EST

    This was a really great exchange and I enjoyed reading it!

    [ Parent ]
    Quality vs Quantity (none / 1) (#196)
    by IAmNos on Mon Jan 10, 2005 at 01:53:34 PM EST

    Wow that was quite a discussion, but I think you both are trying to make it too complicated.  Simple put this becomes an issue of quantity vs. quality.

    Forget the drops of water for a moment.  Lets say I dig a hole  1m x 1m x 1m or one cubic meter.  Now, if I were to dig in the same location and remove another cubic meter, would I have two holes?  No.  The answer is not in the size (quantity) of a hole, but in the definition (quality) of a hole.  A hole is not defined by its size, but by its quality.    

    This becomes the same with the droplets of water.  You can't say adding one droplet of water to another droplet of water makes a particular number of droplets of water.  Instead you can say either that it is still only water, or you can say the volume of water increased.

    I remember Dad doing this little thought trick to me when I was about 10.  Don't look for a complicated answer when a simpler one exists.
    http://thekerrs.ca
    [ Parent ]

    Type safety (none / 1) (#200)
    by curien on Tue Jan 11, 2005 at 01:50:10 AM EST

    What does it mean to "add one droplet of water to another"?

    When I say that I have two apples, I mean that I have two individual apples, perhaps sitting in my lunchbag. I can examine either of the apples, take them apart, eat one but not the other, etc. The point is, the quality "adding" wrt to apples does not destroy either apple's individuality.

    Similarly with numbers, addition is reversible, so 1 + 1 = 2, but also 2 = 1 + 1 and 1 + 1 = 1 + 1.

    Now let's look at the example with the droplets of water (similar reasoning applies to the hole). If I "add" two droplets in the same manner that I added the two apples, I will indeed have two droplets sitting side by side, perhaps on my windshield as I drive to work. I can examine each individually, and maybe my windshield wiper will get one and not the other.

    The complication comes when you perform an operation on the droplets -- combining them to make one larger droplet -- which is impossible with the apples (and indeed, also impossible with numerals). The matter is further confused by deliberately choosing the same word -- ie, "adding" -- to describe this operation despite the fact that it is clearly not isomorphic to our two previous uses of the word.

    This is not a weakness in math, it is a weakness in a particular isomorphism that the narrator has (that is, that combining two droplets of water could use the same operation as adding two apples together). I believe it was gdanjo's point not to describe this as valid but simply to indicate that it is intuitive or, at least, perhaps a commonly-held misconception.

    I agree with gdanjo to a certain extent in that I doubt many people have given it much thought and thus can be easily goaded into giving the wrong answer. If you asked Joe Blow on the street, "Can you add two droplets of water to form a larger droplet?" he'll probably say, "Yes."

    However, as someone else pointed out, the question is quite leading, and doesn't accurately reflect whether he has an innate misconception of mathematical addition. If you asked something less leading, such as, "Would you say that combining smaller droplets into larger droplets is a good analogy for mathematical addition?" you're much more likely to get a negative response.

    --
    This sig is umop apisdn.
    [ Parent ]

    impossible! (none / 0) (#224)
    by efexis on Sat Jan 15, 2005 at 09:49:03 AM EST

    Now, if I were to dig in the same location and remove another cubic meter
    Same location? So you're removing a cubic meter from the hole you've just dug? Yeah I'm being picky, I'll assume you mean you dig another cubic meter NEXT to the location of the previous :-p

    This doesn't really have anything to do with 1+1 though, as you're not adding a hole, you're increasing the size of an existing hole, but this is word play, and I'm being picky still.

    If translating 1+1=2 into holes, or water, it's important to realise that 1+1 doesn't mean add one to the other, it means add both to the context, as sibling poster curien gave as an example, "two droplets sitting side by side, perhaps on my windshield". Yes, more wordplay, but when translating maths into english, you must expect to be subjected to it :-)

    - 2A

    [ Parent ]
    Philosphy sucks (1.66 / 3) (#134)
    by Boronx on Fri Jan 07, 2005 at 05:37:01 PM EST

    This is why philosophy sucks and science rules. There's no way to determine with 100% certainty that the tortoise is actually in front of Achilles, and the level of certainty drops rapidly as the "paradox" wears on.

    What are you going to do when the distance between nose-tips gets to the gamma wave lengths?
    Subspace

    Thank God science is now 100% determenistic... NT (none / 0) (#169)
    by calumny on Sat Jan 08, 2005 at 03:47:15 PM EST



    [ Parent ]
    If we can't measure a fact, is it a fact? (NT) (none / 0) (#182)
    by Boronx on Sun Jan 09, 2005 at 01:03:22 PM EST


    Subspace
    [ Parent ]
    Bad math isn't mysterious (none / 1) (#135)
    by rdmiller3 on Fri Jan 07, 2005 at 05:56:44 PM EST

    Zeno simply neglected to put any actual numbers, or actual math at all, into his "paradox".

    It's reasonable to guess that the reason Plato mentioned that Zeno had no choice in whether to publish (since the manuscript was stolen and published) was that the poor man must have made quite a loud stink about how he wished the theif had NOT published the stuff which he wrote before he learned how to do basic mathematics.

    Out of respect, any decent philosopher should ignore such nonsense.

    Ignorance does not prove an apparent paradox. What was it Ayn Rand used to say, something like, Material reality doesn't have any paradoxes. If you find what looks like a paradox, 'check your premises'.

    ouch (none / 0) (#140)
    by speek on Fri Jan 07, 2005 at 06:20:37 PM EST

    How can one completely miss the point and then say If you find what looks like a paradox, 'check your premises'.?

    --
    al queda is kicking themsleves for not knowing about the levees
    [ Parent ]

    If I get you (none / 1) (#143)
    by levesque on Fri Jan 07, 2005 at 07:15:09 PM EST

    I totally agree.

    First it was an analogy and we all know how hard it is to make a robust one, but to expect any particular one to last over 2000 years, out of context, from hearsay and translated is unreasonable and does not prove anything.

    Second "check your premises" was his point, whether or not you have a paradox, before jumping to conclusions, check your premises. He used paradoxes because he hoped they would act as an incentive to "questioning premises" but this is clearly questionable at this point.

    [ Parent ]

    Proper mathematics (none / 1) (#162)
    by bugmaster on Sat Jan 08, 2005 at 05:12:22 AM EST

    "Proper mathematcis" did not exist in Xeno's time. It took Newton and Leibnitz to get that going. Thus, you can hardly blame Xeno for at least trying, can you ?
    >|<*:=
    [ Parent ]
    Errr.... (none / 0) (#174)
    by porkchop_d_clown on Sat Jan 08, 2005 at 09:50:20 PM EST

    I think you need to consider the period of history we're talking about before you accuse Zeno of neglecting anything. What math did he have at his disposal?

    Should he have worked it out with his handy slide rule? Or perhaps an abacus capable of handling infinite series?

    Has anybody seen my clue? I know I had one when I came in here...
    [ Parent ]

    perhaps the number zero (3.00 / 2) (#177)
    by SIGNOR SPAGHETTI on Sat Jan 08, 2005 at 11:33:51 PM EST


    --
    Stop dreaming and finish your spaghetti.
    [ Parent ]

    Proper Response (1.33 / 3) (#145)
    by Cheetah on Fri Jan 07, 2005 at 07:52:12 PM EST

    It's amazing how many people have not learned / internalized one of the key reactions to dealing with reality: "This is stupid, you are being a dolt, quit bothering me with nonsense." If I can disprove an idea by waving my fingers in front of my face, it really does not deserve any neuron time.

    The only use for Zeno's paradox is to demonstrate how easy it is to construct a fallacy with words when you don't pay attention to what's going on outside the tiny little box on which you've focused. Most of the time people make this mistake in much more subtle ways, Zeno just made up some really ridiculous brain games to demonstrate it clearly.

    Philosophy can be fun, but when it tries to poke at physics, it's usually just being stupid. Philosophy is largely the art of wishing making things so. A cardinal rule of physics, and indeed all science, is that wishing doesn't make it so.

    Xeno's Paradox teaches us something. (none / 0) (#149)
    by Russell Dovey on Fri Jan 07, 2005 at 09:54:39 PM EST

    That is why it is worth speaking of.

    "Blessed are the cracked, for they let in the light." - Spike Milligan
    [ Parent ]

    Calculus (3.00 / 3) (#161)
    by bugmaster on Sat Jan 08, 2005 at 05:11:02 AM EST

    Correct me if I'm wrong, but didn't Xeno basically lay a foundation for calculus with his paradox ? Granted, it's a shallow foundation, but still -- this is a fairly impressive achievement.
    >|<*:=
    [ Parent ]
    Infinite points? (none / 0) (#209)
    by Democratus on Wed Jan 12, 2005 at 09:13:57 AM EST

    I thought Zeno's point was that the world does not behave the same as his paraxox - thus the world must be an illusion.

    Wasn't the paradox supposed to be a proof of the non-existance of the world?

    [ Parent ]

    value of a paradox (3.00 / 2) (#210)
    by tgibbs on Wed Jan 12, 2005 at 04:50:55 PM EST

    The only use for Zeno's paradox is to demonstrate how easy it is to construct a fallacy with words when you don't pay attention to what's going on outside the tiny little box on which you've focused.

    On the contrary, paradoxes such as Zeno's are a good way of testing your understanding--in this case, of basic concepts such as motion or distance. Sometimes, things that seem simple have hidden complexities. You don't "disprove" Zeno's paradox by catching the tortoise; the challenge is to explain why you are able to catch the tortoise.

    [ Parent ]

    Commendation and Questions (none / 0) (#168)
    by calumny on Sat Jan 08, 2005 at 03:42:08 PM EST

    I found the article very thought provoking and well written, but I still have questions about the mathematics used to investigate the paradox (excuse me if this isn't within the scope of your article). It appears to me that Zeno is pointing out the unintuitive nature of infinite summations, especially as applied to physical phenomena, and that an infinite quantity (not potential but actual) has perhaps no physical analog.

    While it is easy to restate the paradox in terms of two linear functions and find their intersection, this does not resolve the original problem, i.e. an infinite number of terms add to a finite sum. It is possible to construe this as an infinitely repeated division of a finite quantity - such as the unlimited number of real numbers between any two integers - but how would one deduce the whole number knowing only the divisions?

    The answer that most everyone seems to be calling for is calculus, and I know that taking the limit of some infinite sequences can give a finite sum, but what is the justification for this technique? Merely that it gives good answers for a set of problems? For a science as rigorous and deductive as mathematics, this seems to undermine the premise of choosing self-evident "truths" for axioms. Or has the theory of limits been neatly axiomatized? It wouldn't surprise me if this advance, like the Hodge Conjecture and many more in the field, is beyond the understanding of all but a few specialists.

    the reason people are talking about calculus (2.33 / 3) (#173)
    by porkchop_d_clown on Sat Jan 08, 2005 at 09:47:28 PM EST

    or, more precisely, integral calculus, is that it was invented to handle these kinds of situations.

    Zeno's paradox can be reduced to a simple statement: that any finite distance can be infinitely subdivided, yet an infinite number of anything shouldn't have a finite sum.

    The second statement is false and using modern math we can easily prove that it is false. But Zeno didn't have those tools at his disposal - so to him it was a paradox.

    To us, it's something to amuse high school sophmores.

    Has anybody seen my clue? I know I had one when I came in here...
    [ Parent ]

    Easily shown (3.00 / 2) (#192)
    by curien on Mon Jan 10, 2005 at 02:26:59 AM EST

    Consider the number 1. It can be represented as a sum of two fractions: 1 = 1/2 + 1/2

    And of course, the number 1/2 can be represented as the sum of two fractions: 1 = 1/2 + 1/4 + 1/4

    That last 1/4 can be represented as the sum of two fractions, too: 1 = 1/2 + 1/4 + 1/8 + 1/8

    Oh, but that last 1/8 can be represented as the sum of two fractions: 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/16.

    Hmm... looks like we have a pattern! We're creating a sequence: a(n) = 1 / 2^n, n>0, so

                  k
                  --
    1 = lim      \   1/2^n
         k->inf  /
                  --
                 n=1

    Wow... an infinite sum, and we're saying it all adds up to 1. Remember, we /created/ the sequence in the first place by examining fractions that all add up to 1. The fact is, that they /still/ add up to 1 -- even when there are infinitely many of them -- because we haven't actually done anything to the numbers that would violate an equation.

    How's that for a justification?

    --
    This sig is umop apisdn.
    [ Parent ]

    Something here doesn't add up (none / 0) (#208)
    by Democratus on Wed Jan 12, 2005 at 09:11:33 AM EST

    None of the example sequences you gave match the formula that you provided.

    All of the examples ended with two identical values (1/2 + 1/2, 1/4 + 1/4, etc.).  Thus you have not shown that your summation is equal to 1 in the same way that your examples are equal to 1.

    [ Parent ]

    No finite sum is (none / 0) (#216)
    by curien on Thu Jan 13, 2005 at 04:37:25 AM EST

    That's the point. The key to working with infinity is to remember that it's not a numer. The only way we can work with it in math is to describe what happens when a number is allowed to increase without bound. That's what the "lim" part says. See what I had to say about that earlier here.

    You're right that if you keep plugging in larger and larger numbers for k, you will get closer and closer to 1 and never actually reach it. But, as long as you're patient enough, you can get as close to 1 as you like even with a finite sum.

    And that's all that dealing with infinites really means -- you can get as large as you want, as small as you want, approach a value asymptotically etc as long as you have patience.

    If Zeno's paradox disproves the existence of anything, it's not reality -- it's the existence of infinity as "a number" that can be conceived like other numbers.

    Consider this. The number 1 can be represented as 1, or 1+0, or 1+0+0, or 1+0+0+0, or 1+0+0+0+0, etc. In fact, you could write it with an "infinite number" of plus zeros. So what? It still equals 1, right? How's that, though? According to Zeno, in order to reduce it to 1, you'd have to add zero an infinite number of times! And since you can't do that, addition must not exist.

    All Zeno showed is that he had an incomplete understanding of infinity. (Which isn't surprising, considering they didn't even understand 0 yet.)

    --
    This sig is umop apisdn.
    [ Parent ]

    Better justification (none / 0) (#239)
    by curien on Tue Jan 18, 2005 at 09:34:29 AM EST

    First, let's make things a little more general:

    N = 1/A + 1/A^2 + 1/A^3 + ...

    Now, multiply through by A
    NA = 1 + 1/A + 1/A^2 + 1/A^3 + ...

    Do some arithmetic:
    NA - 1 = 1/A + 1/A^2 + 1/A^3 + ...

    Now we've got the original sequence back, let's substitute in N.
    NA - 1 = N

    Solve for N
    N(A - 1) - 1 = 0
    N = 1 / (A - 1)

    So, it follows that
    1/(A-1) = 1/A +1/A^2 + 1/A^3 + 1/A^4 + ...

    In my example sequence earlier, A was 2, and 1/(2-1) is indeed 1. Note that this is argument has the exact same form as the one used to show that .3333... is the same as 1/3.

    --
    This sig is umop apisdn.
    [ Parent ]

    Partially valid criticism (none / 1) (#203)
    by epepke on Tue Jan 11, 2005 at 12:50:54 PM EST

    Eudoxus, around 400 BC, has a rough version of calculus. Leibnitz and Newton are usually credited with having independently invented an essentially modern form.

    However, at this point, calculus didn't really have a sound mathematical footing. That had to wait for Fermat and some later mathematicians. The good news is that it is sound, and it is solid for limits. The bad news is that you are right; it takes a lot of work to understand, a couple of orders of magnitude more work than just doing calculus.


    The truth may be out there, but lies are inside your head.--Terry Pratchett


    [ Parent ]
    Compact Sets (3.00 / 2) (#204)
    by HKDD on Tue Jan 11, 2005 at 08:47:35 PM EST

    Or has the theory of limits been neatly axiomatized?
    Yes.

    Compact sets solve these problems of adding infinite  numbers of things.  

    Since we believe that physical space behaves like the reals in 3 dimensions, and each of Zeno's paradoxes "takes place" in a closed, bounded subset of real space(ie, a race, a stadium, etc), obviously the actions he describes will necessarily converge.  Thus motion is possible, and the faster runner can win the race.

    It has very little to do with any intuitive notions of limits, infinity, and divisibility.  The mathematical argument has everything to do with the strictly mathematical concepts of set, closedness, boundedness, sequence, and limit.  And I agree with you, that these are things that someone who took "calc" in college won't understand well.  Because everyone and their brother takes calc in college, from engineers to business to doctors.  So calc is used as a weed out class where you don't end up understanding anything, but you can do the calculations.  Probaly none of them could rigorously prove the so-called "Fundamental Theorem" of calculus.  But proofs aren't really that difficult, just not studied.

    So if you take the logical, rigorous definitions of set, closed set, bounded set, sequence and limit and analyse any of Zeno's paradoxes(where he will describe a sequence of events, located in a compact subset of reality), you will see that they are easily solvable.  You will always find a convergent sub-sequence.  

    Hell you don't even have to get all philosophical on the nature of "motion":  is it a change in position over time? or is it a completion of successive steps?  I think this is where people get confused, especially concerning Zeno.  Because they think that they actually understand "motion" because they "took calculus."

    But on the flip side of things, there is NO rule that says you must agree that physical space behaves like the real numbers in 3 dimensions.  There is also NO rule that, even if you believe that physical space is like the reals, that the situations Zeno describes are compact sets.  And still yet, there is NO rule that, even if you agree that physical space is like the reals and Zeno is describing a compact set, that we have actually found a convergent sub-sequence.

    It really is a stricly logical paradox, though not a mathematical or physical one.

    You can't escape the danger!
    [ Parent ]

    A lead (none / 0) (#205)
    by calumny on Wed Jan 12, 2005 at 01:19:01 AM EST

    I was floundering around the definition of limit in analysis and was little satisfied. Everything that I found on limits and convergence made it abundantly clear that the infinitesimals of Newton and Leibniz were poorly understood and widely disparaged, but I could never find any talk of an alternative. Except for a few lines of "finally, calculus was placed on a theoretical basis sometime in the twentieth century" (how?), I was getting no where. Thank you for pointing me in a new direction.

    [ Parent ]
    Why the 1 rating? (none / 0) (#206)
    by curien on Wed Jan 12, 2005 at 01:36:06 AM EST

    Just curious. If there's a part of my explanation you don't understand or don't feel is justified, let me know, and I'll try to improve it.

    --
    This sig is umop apisdn.
    [ Parent ]
    My misjudgment (none / 0) (#211)
    by calumny on Wed Jan 12, 2005 at 04:51:48 PM EST

    No offense intended. Upon first reading it, I noticed that while it demonstrates a special case, it didn't seem rigorous enough to convince me that there is an entire set of sequences that sum to a constant. Also, it doesn't obviate the need for infinitesimals, as it still calls for the summation of an infinite number of infinitely small terms. (Or, as someone pointed out, infinitesimals can be well defined - as they are in nonstandard analysis.)

    However, seeing it as a clarification of the paradox is valuable. As a description of calculus - especially Archimedes' ancient analog in the method of exhaustion - it is very intuitive. I ratcheted the score up in light of this and the extra investigation it inspired.

    [ Parent ]

    OK, thanks (none / 0) (#213)
    by curien on Thu Jan 13, 2005 at 01:44:29 AM EST

    Mostly, I just wanted to make sure you didn't think I was trying to be a smart-ass or something.

    No, it wasn't really meant to be rigorous. I guess I misguessed how much math you could handle and ended up going for a more "intuitive" explanation.

    --
    This sig is umop apisdn.
    [ Parent ]

    I've never been able to understand (none / 1) (#186)
    by lastobelus on Sun Jan 09, 2005 at 07:23:33 PM EST

    why this is a paradox. Every time he checks their relative positions, he chooses a time to check that's in between the last time he checked and the time at which Achilles will pass the tortoise. So he never sees Achilles overtaking. Obviously. How is it a paradox? I never understood what is supposed to be paradoxical about it and it always made me feel so stupid.

    Assumptions (none / 0) (#193)
    by curien on Mon Jan 10, 2005 at 02:50:12 AM EST

    The paradox is meant to appeal to the idea that anything requiring an infinite number of steps can never actually be accomplished. Zeno took that as an axiom, and postulated that anything that can be stated in such a way as to require an infinite number of steps can likewise never be accomplished.

    In some circumstances, this is true. If an object is infinitely far away, each (literal) step you take toward the object does not take you closer to it -- it's still just as far away as it was before; that is, it's still infinitely far away. eg, Which is closer to infinity -- 1 or 2?

    In other cases, as with the race, his axiom conflicts with reality.

    What really kills his postulate wasn't even understood until the 19th C (cf Cantor diagonals): that is, that some infinities are "more infinite" than others.

    The real kicker is that in the general case, Zeno's postulate is true: Most algorithms that require an infinite number of steps cannot be accomplished in a finite amount of time. There's only a small (well, "less infinite") set of such algorithms that violate the postulate. The paradox describes one element of that set, but Zeno didn't account for the special case.

    --
    This sig is umop apisdn.
    [ Parent ]

    Re: Assumptions (none / 0) (#217)
    by Kalani on Thu Jan 13, 2005 at 01:06:07 PM EST

    What really kills his postulate wasn't even understood until the 19th C (cf Cantor diagonals): that is, that some infinities are "more infinite" than others.
    How do Cantor's transfinite cardinals destroy Zeno's Paradox? I've always interpreted Cantor's work as a more rigorous and formal description of Zeno.
    The real kicker is that in the general case, Zeno's postulate is true: Most algorithms that require an infinite number of steps cannot be accomplished in a finite amount of time. There's only a small (well, "less infinite") set of such algorithms that violate the postulate. The paradox describes one element of that set, but Zeno didn't account for the special case.
    But don't you actually mean that some infinite algorithms can be reduced to finite ones? Clearly an algorithm that's actually infinite won't ever halt.

    I think that Zeno's Paradox is an argument against continuous space/time/matter. At some point, a quantum of matter must jump a quantum of distance in a quantum of time (ignoring questions about the topology of space).

    -----
    "I have often made the hypothesis that ultimately physics will not require a mathematical statement; in the end the machinery will be revealed
    [ Parent ]
    no, it still isn't a paradox (none / 0) (#207)
    by exa on Wed Jan 12, 2005 at 03:47:09 AM EST

    it's not true in any nomologically possible world. you don't need to use mathematics to show that this particular paradox is not a physical paradox, e.g. there is no such contradiction in a physical system.

    this is not the most interesting paradox of zeno, anyway. i don't understand why people are keen on this paradox.

    a few years ago i criticized on sci.physics, hopefully fatally, an independent researcher Lynds who managed to get his misunderstandings of Zeno published in a physics journal. i don't want to sound dismissive, but I found the section Aliens and the Art of Boxing and the stuff after that  highly irrelevant with respect to this particular paradox.

    some paradoxes attributed to zeno are indeed interesting, but some of them do not give rise to either logical or semantic antinomies. the one you wrote about is one of them, apparently.

    also, many authors regard zeno as the first mathematician, while others regard pythagoras as the first mathematician. i've seen a reference to zeno in many historical treatises of mathematics. so, it's no exaggeration to say that...

    Regards,

    __
    exa a.k.a Eray Ozkural
    There is no perfect circle.

    never was a REAL paradox (none / 0) (#212)
    by gdanjo on Thu Jan 13, 2005 at 12:22:00 AM EST

    it's not true in any nomologically possible world. you don't need to use mathematics to show that this particular paradox is not a physical paradox, e.g. there is no such contradiction in a physical system.
    You're right, we don't need mathematics to see that the paradox isn't physical - after all, I can overtake cars while I drive.

    The paradox is in our intuitive model of the world, and in trying to reconcile what intuition would lead you to believe and what reality tells you to believe.

    a few years ago i criticized on sci.physics, hopefully fatally, an independent researcher Lynds who managed to get his misunderstandings of Zeno published in a physics journal.
    It is Lynds' attempt at a solution to the arrow paradox that I link to in the introduction. He's attempting a physical solution to the paradox - I'm not. I'm just trying to understand the model that we use to describe it.

    I do think Lynds' ideas are interesting though, and I wouldn't mind hearing your critisisms of them.

    i don't want to sound dismissive, but I found the section Aliens and the Art of Boxing and the stuff after that highly irrelevant with respect to this particular paradox.
    It's only relevant as an allegory to better understand why a mathematical solution is sufficient for mathematicians but not the layperson - just as a contemporary description of boxing to an alien is insufficient for they most likely lack concepts required for such a description to be meaningful.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    Lynds (none / 0) (#225)
    by exa on Sat Jan 15, 2005 at 10:41:49 AM EST

    I read Lynds's paper and I think it was terrible. If you search for my name, Eray Ozkural, on sci.physics.* you will see my criticism. It was unfortunately a bitter criticism, but a well deserved one.

    Regards,
    __
    exa a.k.a Eray Ozkural
    There is no perfect circle.

    [ Parent ]

    Infinite confusion (none / 1) (#218)
    by acronos on Thu Jan 13, 2005 at 06:15:31 PM EST

    This continues on forever. Therefore Achilles never overtakes the tortoise.

    This statement is where he pulls the hoodwink in the paradox. He converts what all will agree is an infinite number of steps into an infinite amount of time. "This continues forever." should be "this continues for an infinite number of steps." However, it does not have to take an infinite amount of time to cross an infinite number of steps. Whether it does or not will depend on the size of the steps. If they tend toward the infinitely small we get something like.

    2 seconds + 1 second + 1/2 second + 1/4 second .... on to infinity

    and when the time to take all these steps are added up you will have 4 seconds not infinity.

    actually (none / 0) (#231)
    by gdanjo on Mon Jan 17, 2005 at 05:23:33 AM EST

    "This continues forever." should be "this continues for an infinite number of steps."
    Actually, it should be "this potentially continues for an infinite number of steps" - if it actually did continue for an infinite amount of steps it would actually take forever. In reality, the universe takes a short-cut somewhere (likely at the quantum level) to prevent this (potential) impass.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    I wonder. (none / 0) (#233)
    by ciphertext on Mon Jan 17, 2005 at 11:39:52 AM EST

    That is a true statement if either or both of these conditions are true; your infinite steps produce an infinite sum or, your speed was reducing in proportion to the distance you travel with each step you take.

    Obviously, if you are unfettered with an end, then you can produce as many steps as you want and not get to where you are going. Similarly, if your rate of speed decreases in proportion to the increase in distance you travel, then you will never get to where you are going.

    I liked your philosophical musings. They are fun to think about. Sort of like a jog around the track for your mind.



    [ Parent ]
    This is not true (none / 0) (#236)
    by acronos on Tue Jan 18, 2005 at 12:34:35 AM EST

    if it actually did continue for an infinite amount of steps it would actually take forever.

    No, it wouldn't. If you want proof wouldn't, study limits and calculus which are both continuous not discrete mathematics. Or just take a breath; the amount of time required to take that breath can be divided indefinitely so the breath you just took required an infinite number of instances yet only took a finite amount of time. It is quite possible for an infinite number of finite steps to result in a finite number.

    [ Parent ]

    ugh (none / 0) (#242)
    by gdanjo on Tue Jan 18, 2005 at 06:51:40 PM EST

    Or just take a breath; the amount of time required to take that breath can be divided indefinitely so the breath you just took required an infinite number of instances yet only took a finite amount of time.
    The key phrase here is can, but if you go ahead and divide that time into an infinite number of "instances", it will take you forever.

    I understand calculus, thank you very much, but it seems you don't understand the difference between "could" and "did."

    It is quite possible for an infinite number of finite steps to result in a finite number.
    Wrong. It's possible divide a finite number any number of times - this number of times can be as large as you want, but NOT "infinite" in the literal sense. It's also possible to estimate (quite accurately) a limit of a potentially infinite amount of steps, but nobody - I repeat, NOBODY - is able to sum up a truly Infinite number of steps.

    There is a difference between the mathematical notion of infinity and Infinity in the literal sense. And even though this difference may not matter - may never matter - you can't deny that there is a difference.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    Differences (none / 0) (#243)
    by acronos on Tue Jan 18, 2005 at 09:54:20 PM EST

    I repeat, NOBODY - is able to sum up a truly Infinite number of steps.

    There is a difference between the mathematical notion of infinity and Infinity in the literal sense. And even though this difference may not matter - may never matter - you can't deny that there is a difference.

    Plenty of people are able to sum up an infinite series. While I agree that it would take a forever to sum such a series manually, that is like saying it is impossible to count to 100 because you have limited yourself to using your fingers and toes. There is a difference between your notion of infinity as expressed here and reality and fundamental calculus. In reality the series I iterated in my first post would take 4 seconds to complete, and it would not matter whether atomic theory was continuous or discrete. If you start adding that series, it is easy to see that you will NEVER get a number greater than 4 seconds. The correct answer is NOT infinity. That you think so shows that you do not understand calculus, limits, or infinity.

    This statement: It is quite possible for an infinite number of finite steps to result in a finite number. is pretty much the fundamental concept in calculus.

    The only thing I can think of that might cause you not to get this is a confusion between how long it would take you to calculate such a series manually vs. how long it would take the series to happen in reality. While, your ability to calculate it manually would require a static amount of time for each iteration, when the system actually happened in reality each iteration would get smaller and smaller until it was infinitely small - 1/infinity.

    [ Parent ]

    consider this proposition: (none / 0) (#244)
    by gdanjo on Wed Jan 19, 2005 at 02:45:11 AM EST

    Is it possible to represent an infinite amount of terms with a finite (single) label? For example, can I represent the symbol '1' as the sum of an infiniite series?

    Well, we can say 1 = 1/2 + 1/2, and further, 1/2 can be represented by 1/4 + 1/4, and so on, ad infinitum. Therefore, the answer is yes, we can represent an infinite series with a finite symbol.

    But wait! We have just assumed the thesis: we assume that 1/4 can be represented by an infinite series. In fact, whenever you stop and declare the answer to be 'yes', you imply that the last finite symbol presented represents an infinite series.

    Now, it's easy to say that the thesis is proven, for no matter how many terms you specify, you cannot disprove it, but you cannot get away from the fact that at some point you need to assume the thesis for the answer to be 'yes.'

    In mathematics this is good enough to be called "infinity", but as Heath says, this is a different notion of infinity than the literal one. It's probably good enough for all but philosophical definitions, but that's my point! There exists a definition (the literal one) that does not fit the mathematics.

    The only thing I can think of that might cause you not to get this is a confusion between how long it would take you to calculate such a series manually vs. how long it would take the series to happen in reality.
    How long it takes for a "series to happen" in reality is just a better estimation than a manual calculation. There's no reason to think that space is infinitely divisible (ever heard of quantum mechanics?), so the confusion here is yours.

    Again, I think you miss my point. Read the Heath quote in my article; mathematics falls apart when it tries to deal with literal infinities. The two are different, and mathematics has let go of the literal definition, content with it's approximation ("as large as you like") since it is "good enough" for all but philosophical definitions.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    How many angels can dance on the head of a pin (none / 0) (#245)
    by acronos on Wed Jan 19, 2005 at 11:14:47 AM EST

    This is like trying to decide how many angels can dance on the head of a pin. It is totally useless to make such distinctions between different infinities. This is especially played out when your assumptions create incongruence between math and the natural world. The whole point of math is to model the natural world. It is the natural world that is the source.

    The entire Zeno's paradox is a mathematical concoction based on outmoded and obsolete notions of infinity and zero. In the absence of math, the paradox doesn't exist. In modern math, the paradox doesn't exist. It is only in ancient concepts of math that the paradox exists. We don't seem to be making any progress. I see no value whatsoever in your argument.

    I can't help myself from addressing one more point.

    There's no reason to think that space is infinitely divisible (ever heard of quantum mechanics?), so the confusion here is yours.

    I am well aware and agree with the notion that space is quantized. My point was that it would not matter whether it was or not because Zeno's paradox is invalid either way.

    [ Parent ]

    questions (none / 0) (#247)
    by gdanjo on Wed Jan 19, 2005 at 07:54:28 PM EST

    This is like trying to decide how many angels can dance on the head of a pin. It is totally useless to make such distinctions between different infinities.
    Of course it's useless, when you've defined 'useful' to mean 'mathematically useful.' The traditional notion of the literal infinity has been banished from mathematics - so it's useless to all but philosophers.

    However, you have finally admitted your disdain for philosophy, so we can end the conversation right here. You need not convince me of the usefulness of the mathematical notion of infinity, but given your mathematically biased judgement of usefulness, I need not continue to try and convince you of the meaning of a literal infinity. Which was, I thought, the point having a philosophical discussion.

    I will end here with the observation that you didn't even attempt to answer my question in the previous post, presumably because you believe the question to be 'useless.'

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    I did try to answer your question. (none / 0) (#252)
    by acronos on Sat Jan 22, 2005 at 05:58:53 PM EST

    I thought my last post would be the last.  However, I want you to know that I do appreciate your efforts to expand my thinking, and I usually do find philosophical discussions interesting.  Something gets me about Zeno's paradox, because most people see a paradox, but I don't.  I think you had in mind a fun conversation, but I got frustrated that everyone else couldn't see what was so incredibly obvious to me.  I repent.

    Math and logic are human creations.  They are human creations created specifically to model reality.  When the math and logic that we use don't match reality, then it is not reality that I question - it is the math and logic.  When the conclusion of some mathematical concoction is that reality is impossible, then my conclusion is that that is BS.  Especially when the math that I tend to use doesn't create such contradictions.

    I respect your opinion.  But I respectfully believe that your opinion is based on math and logic that are very old.  Math has come a long way in the last 2000 years.  I think of the paradox you stated as causing the distance between Achilles and the tortoise to get shorter and shorter until it eventually becomes zero and doing so at a faster and faster rate until it eventually becomes infinity.  Eventually in both cases is 4 seconds in my earlier example.

    Mathematics developed tools to deal with the paradox created around infinity.  Those tools were limits and calculus.  The tools accurately model reality.  So I guess you are right, because as long as it works, it's good enough for me.  I do not see the paradox.

    I understand that you are trying to distinguish between a good enough infinity and a real infinity.  However, I don't agree with this distinction.  In a limit, it is real infinity that is being approached.  While your argument is that we never arrive, I don't think that is significant because the answer we get out of the limit is the correct one as represented by reality.  If it were not, calculus would not work.

    As time passes we will get better and better tools for thinking about infinity.  As we do, students trained in such math and logic will find less and less value in such a paradox because they will have the tools to see it is not a paradox.  It is only a paradox when you have just enough intellectual tools to conceive of the problem but not enough to see the solution.  It is a problem in old math and logic, not reality.  It shows the glaring hole in our tools, not the hole in reality.


    [ Parent ]

    Is space quantized? (none / 0) (#257)
    by coopex on Fri Apr 08, 2005 at 04:15:44 PM EST

    I mean, it's scientifically proven that energy is, but I've never heard or seen the same about time or space.  If you have a link, paper title, or book name I'd be most grateful.

    [ Parent ]
    Logical Positivism (none / 0) (#249)
    by HKDD on Wed Jan 19, 2005 at 10:37:55 PM EST

    I understand what you're saying.  I agree that it is a logical paradox; you must assume the thesis and all that.  I get it.

    However, isn't a "literal definition of Infinity" nonsensical?  How would a philosopher use such a thing?

    The only reason I ask is that I don't understand how the mathematical definitions are "approximations" when considered by a philosopher.  Mathematical truths are tautologies; analytic propositions.  What synthetic propositions would make use of a "literal Infinity"?
    You can't escape the danger!
    [ Parent ]

    because (none / 0) (#250)
    by gdanjo on Thu Jan 20, 2005 at 05:41:13 AM EST

    However, isn't a "literal definition of Infinity" nonsensical? How would a philosopher use such a thing?
    So you're asking a) whether a literal definition of infinity makes sense, and b) whether a literal definition of infinity is useful.

    To the first question, I answer "yes" - because the literal definition is, I would argue, more intuitive than the mathematical definition, which is basically "as big as you want", and therefore weaker than the literal definition (which is "bigger than you [can] think").

    As for it's usefulness, well, mathematics has deemed it not very useful at all; it fucks with the rest of mathematics, and so has been banished. But it is still useful as a concept in-and-of-itself, since a) it can be defined, and b) it can be understood (you, for example, seem to understand it).

    It is philsophically useful because "it's there"; it can't be denied. Philosophy, unlike mathematics, has a far weaker requirement of interest when it comes to consideration of concepts: mathematics requires rigorous proofs, or to be so useful as to be elevated to an axiom; philosophy only requires that it can be communicated (defined; understood).

    And to show one example of where a literal definition is useful, re-read the beginnings of this very thread, where statements such as "this continues for an infinite amount of steps" can be clearly shown to be false.

    Once again I must stress that it is the realm of philosophy that I am interested in, and in no way do I try to diminish the mathematical definition. I'm just asserting that they're different, and we must recognise when a literal definition is used in natural language so we don't say stupid things that mathematics has already rejected.

    The only reason I ask is that I don't understand how the mathematical definitions are "approximations" when considered by a philosopher.
    I call a mathematical definition an approximation only because it is "weaker" than the literal definition. It's an approximation only in that we cannot instantiate an infinite amount of terms, but I do not mean that the result is somehow less correct than if we actually were able to sum a literally infinite set of terms; it's just an admission that we take a shortcut, though it most probably results in a most accurate answer.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    Easy way to sum an infinite series using algebra (none / 0) (#256)
    by coopex on Fri Apr 08, 2005 at 04:03:22 PM EST

    Sum = 1 + 1/2 + 1/4 + 1/8 + ...
        = Sigma( n = 0 to infinity ) 1/2^n
    So 2*Sum = 2 + 1 + 1/2 + 1/4 + 1/8 + ...
      2*Sum = 2 + 1 + 1/2 + 1/4 + 1/8 + ...
     -Sum = 1 + 1/2 + 1/4 + 1/8 + ...
     =Sum = 2

    Wow, I just proved in half the space of this comment box what took you 4 screens.

    [ Parent ]

    Might want to keep your mouth shut... (none / 0) (#255)
    by coopex on Fri Apr 08, 2005 at 03:55:23 PM EST

    about stuff you know jack about, just makes you look stupid.

    If you understand calculus, you would understand that infinite series can have a finite sum, you are evn given the formula for the sum of an infinite geometric series in algebra.  

    Go spend some time away from you computer, at a place called a library, read some remedial math books, realize that you're a moron for trying to use English to do math and learn do math problems using the language of math

    [ Parent ]

    what's the big deal (none / 0) (#219)
    by klem on Thu Jan 13, 2005 at 11:07:52 PM EST

    i thought calculus, by introducing the idea of summing inifinitely small things to get a finite thing, basically removes the paradox from the achilles and the tortoise story

    Hmm.. (none / 0) (#220)
    by spooky wookie on Fri Jan 14, 2005 at 07:14:22 AM EST

    "Even now it would be a brave call to declare the paradoxes resolved."

    Seriously? It seems quite obvious even to an lay-man idiot like myself that any real world experiment involving A well trained runner and a tortoise would debunk this "paradoks". Just because you can do something mathematically doesn't mean it applies to reality.

    no shit (none / 0) (#230)
    by gdanjo on Mon Jan 17, 2005 at 05:19:48 AM EST

    sherlock. The paradox does NOT apply to reality, as any real person can attest.

    The paradox attacks a fundamental aspect of knowledge, and I'm saying that just as other knowledge-based paradoxes are unlikely to ever be solved (eg: Sorites' paradox), neither will this one.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    Quantum mechanics? (none / 0) (#222)
    by kreyg on Fri Jan 14, 2005 at 04:27:41 PM EST

    I had considered at one time that due to energy not being continuous, the infinite halving is not physically possible. Time is not continuous, so at some point there is a minimum amount you can move in a single instant.

    Has this ever been considered before? Based on this, it seems to me the entire premise of the "paradox" is invalid.


    There was a point to this story, but it has temporarily escaped the chronicler's mind. - Douglas Adams

    weak (none / 0) (#223)
    by balsamic vinigga on Sat Jan 15, 2005 at 05:02:15 AM EST

    weak ass argument.  Achilles lapping the tortoise, be it on a race track, the earth, or some other circular course doesn't make him behind the tortoise.  Your assertion that it does practically makes no sense at all.  You tyr to be so smart in this article, it's funny to see you struggling with such a fundamental confusion of distance to whose ass is facing who.  Suppose they were racing by running backwards..  does that make the faster runner "behand" the slower runner?  The shit you postulate is equally absurd.

    ---
    Please help fund a Filipino Horror Movie. It's been in limbo since 2007 due to lack of funding. Please donate today!
    yuo misunderstand (none / 0) (#229)
    by gdanjo on Mon Jan 17, 2005 at 05:16:46 AM EST

    Achilles lapping the tortoise, be it on a race track, the earth, or some other circular course doesn't make him behind the tortoise. Your assertion that it does practically makes no sense at all.
    It's not a question of who's in front of whom, it's a question of overtaking. Zeno's conclusion is NOT "the tortoise wins the race", it's "Achilles never overtakes the tortoise" - as in "Achilles never goes from immediately-behind to immediately-in-front-of the tortoise."

    It's an important distinction because in the first fable, the intuitive conclusion is that the tortoise wins, which is false - in the inverted second fable the intuitive conclusion (that the tortoise and Achilles never "switch positions", and therefore Achilles "wins") is once again that no "switch" occurs, which is also false; they're both false in the same way.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    whatever (none / 0) (#240)
    by balsamic vinigga on Tue Jan 18, 2005 at 12:49:37 PM EST

    Again, do you not know what it means to overtake someone in a race?  If you lap someone you're not overtaking them again..  it means you overtook them a lap ago.  Races are measured in distance, not your position along a linear path.  Zeno's real paradox aknowledges this.  This attempt at making some variation or a new paradox just looks absurd.

    ---
    Please help fund a Filipino Horror Movie. It's been in limbo since 2007 due to lack of funding. Please donate today!
    [ Parent ]
    point, miss thyself (none / 0) (#241)
    by gdanjo on Tue Jan 18, 2005 at 06:43:50 PM EST

    A race is a virtual construct that specifies criteria on the rules of the contest, and judges a winner based on this criteria. Zeno's paradox is about the impossibility of motion - specifically, the impossibility of a faster moving object overtaking a slower moving object.

    I repeat: it is NOT a race in the way you seem to think it is.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    What the hell? (none / 0) (#246)
    by balsamic vinigga on Wed Jan 19, 2005 at 11:16:41 AM EST

    I understood zeno's paradox back in first semester calculus.. Don't be handing out this condescending bullshit.  Yeah ok so it assumes all Series add up to inifinit, which we now "know" better and thus it breaks.  But my point isn THAT the whole "the tortoise gets lapped and thus is never overcome" bullshit is such a stretch and pointless waste of speculation.

    ---
    Please help fund a Filipino Horror Movie. It's been in limbo since 2007 due to lack of funding. Please donate today!
    [ Parent ]
    the whole point (none / 0) (#248)
    by gdanjo on Wed Jan 19, 2005 at 08:14:20 PM EST

    of the article is to get you, the reader, to think of the fable in different terms than the way in which you normally think of it - as a mathematical proposition, with a mathematical solution. There's a reason the fable has lasted so long, and the reason is that it panders to the intuition. Thus, for me, an intuitional solution is the one that will be most convincing, especially if you're not trained in the language of mathematics.

    Now, given this, why is my description of the race 'invalid'? It's ok if you don't want to think about my analysis, just don't reply to this post and it'll be over. But for you to say I make no sense requires that you understand what I'm saying.

    I don't mean to get narky, it's just my style. Given your nick, I would have assumed you were used to a rough'n'tumble style of conversation, and have thicker skin to protect you from perceived condescention.

    Dan ...
    "Death - oh! fair and `guiling copesmate Death!
    Be not a malais'd beggar; claim this bloody jester!"
    -ToT
    [ Parent ]

    This is why philosophy is BS (none / 0) (#254)
    by coopex on Fri Apr 08, 2005 at 03:44:46 PM EST

    For every tricky problem there is an answer that is simple, intuitive and WRONG.

    There's a reason that mathematics and science developed their own precise and terse language, so that you could PROVE, not just convince someone, something in two lines which took you 100 times that to just make some hand waving arguement.  Aristotle asserted that heavier objects fall faster than lighter ones, while Galileo actually did an experiment, and PROVED as far as anything can be proved non-mathematically that Aristotle completely wrong.

    There's a reason mathematical problems have mathematical solutions - because that's the entire reason math was developed, to solve math problems.  Trying to solve math problems in English is just as stupid as trying to write an essay on MacBeth in predicate calculs.

    [ Parent ]

    Greatest comment ever: (none / 0) (#261)
    by An onymous Coward on Mon May 02, 2005 at 07:00:13 AM EST

    "But for you to say I make no sense requires that you understand what I'm saying."

    "Your voice is irrelevant. Stop embarrassing yourself. Please." -stuaart
    [ Parent ]
    Something + something = something else (none / 0) (#251)
    by ZenRock on Fri Jan 21, 2005 at 12:57:14 PM EST

    The REAL distance between Achilles and the Turtle is not divided in any way. Points do not exist in reality. They are only a crutch our feeble minds must use in order to break reality up into small enough pieces to marginally understand it. But the real space between them is perfectly smooth and without ANY division. The problem is with the mathematical model, not reality.

    First we divide the space between 2 objects in order to try and measure the distance. Then we increase the amount of points to try and smooth out the space to more closely imitate the reality of it. But we can never put enough points into the space to do that because it never had any division points to begin with. It's perfectly smooth and continuous. The only way to smooth the mathematical model completely is to stop using points altogether.

    The other problem is getting caught up with this fudge word called "INFINITY" The dictionary definitions of the word infinity are very revealing. "An indefinitely large number or amount" or "Unbounded space, time, or quantity" It simply boils down to a question mark - an unknown - undefined amount. So it stands to reason if you add and undefined amount to another undefined amount you still have an undefined amount. Although you know it must be a different amount you don't know how much different.

    If you add something you don't know to something you don't know, the answer is
    - you still don't know!!

    Even if you add a known amount to an unknown amount the answer is still an unknown amount.

    Infinity + 1 = Infinity
    Question mark + 1 = Question mark

    Infinity^2 = Infinity
    Unknown amount^2= Unknown amount


    Why does E ALWAYS equal MC squared?

    but (none / 0) (#262)
    by keleyu on Thu May 05, 2005 at 09:08:26 PM EST

    But such a conclusion would be premature - where on earth could you have such a race where the evaluation part of this story is true forever?

    Deconstructing Infinity: An Analysis of Zeno's Paradox | 262 comments (207 topical, 55 editorial, 0 hidden)
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