Saturday, January 08, 2005

Deconstructing Infinity

An Analysis of Zeno's Paradox.

...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.


Blogger The Doctor said...

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.

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 ..."

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."

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 naïve concept of infinity.

And for that, he will be remembered forever.

6:52 PM  
Blogger The Doctor said...

From the comments:

"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.

6:57 PM  

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