[Thomas An.]

Enough slouching around fellows ... time for some more thinking

It seems that the human mind is simply not wired to comprehend infinity. The mere presence of that notion produces brain teasers

For example:
An ancient philosopher by the name Zeno of Elea formed the following argument...

• Imagine a fast runner by the name Achilles
• Imagine a slow running turtoise
• Give the tortoise a head start (say 1000 meters) at a point A1
• Both runners start and maintain a constant speed
• When Achilles finally reaches point A1 the tortoise has already moved to a point A2.
• When Achilles finally reaches point A2 the tortoise has already moved to a point A3 ... and so on ad-infinitum
• Thus in theory Achilles (a fast runner) can never overcome a tortoise (a slow runner)

Falacy detection:

• 
  Suppose we introduce an imaginary observer in the Achilles/Tortoise scene.
• The observer's task is to place a mark (a flag) to the current position of the Tortoise every time Achilles manages to reach its previous point.
• When Achilles reaches point A1, the observer moves the flag to point A2, and then to point A3 .... and so on.
• The above process entails that the observer will need to act progressively faster. Achilles will reach point A1 in some time t, but then he will reach point A2 in a shorter time.
• The observer's speed theoretically, will need to keep increasing to meet the task ... passing the speed of sound, eventually passing the speed of light and then approach infinity.
• The need for this observer to maintain a near infinite speed will require to consume enormous amount of energy and eventually all the energy stored in the entire cosmos. At that point theoretically, the cosmos ceases to exist (including the observer himself, Achiles and Tortoise). In that sense the observer never witnesses the event of Achilles meeting the Tortoise.

Upholding the initial argument involves the theoretical destruction/consumption of the known universe (or a similar catastrophic/degenerative event). So the initial argument, when fully stated, becomes: "We the observers" will never see Achiles overtake a turtoise as long as our observation speed becomes infinite... at which point time has no meaning".

The moment we interpret Achiles objesctive as merely an attempt to reach the next flag then we are deliberately setting the "meeting event" to be an assymptotic barrier. In other words, an observer that looks for Acheles to meet the turtle's previous position, is an observer that refuses to see them meet and will do anyhting (move at infinite speeds) to prevent that from happening.

As such, I would argue that the true fallacy in this paradox is that it requires infinite observer speed which would entail consumption of the universe in an attempt to pause time. (Any such requirement for infite observer speed and pausation of time "usually" implies a fallacy in the premise of the initial argument )

P.S. I feel the standard method that math teachers use in calculus classes, by means of integration, is philosophicaly and fundamentally flawed.

[Mihai]

It seems that the human mind is simply not wired to comprehend infinity.

"Comprehend" is the wrong choice of word IMO. Is anything really infinite? It's just a practical word describing a scale relating to our own being. We put "infinite" figures to help describe a scale which to us, seems infinite, but is it really? 1 000 000 000 trillion years would seem pretty infinite to us, same with energy.

An ancient philosopher by the name Zeno

He should just make a little better Gedankenexperiment no?

[jdp]

Thomas, as far as I know the paradox is nonetheless mathematically solved with the introduction of calculus (leibniz) unless the philosophical meaning of it, which it's still unsolved and disputed.

he took us an entire night, an year or so ago, debating with my roommates about it; we were able to find some solution on the internet demonstrating that the series is essentially of this type:

1/2+1/4+1/8+1/16+..., which is a limit converging to 1 (not to infinite)

here's what I was able to find out in english regarding the math behind it.

bertrand russel wrote something interesting about it: "if Achilles will ever catch the turtoise, this must happens only after an infinite set of instants since its start. This as a matter of fact is true: what it is not true is the fact that an infinite amount of instants gives an infinite time, thus it's impossible to say that achilles won't ever catch the turtoise" (sorry for the bad translation)

[victor]

There is an equivalent story with an arrow, isn't it? For the arrow to fly away 1 meter, it needs to reach 0.5 meter first, and 0.25m first, and 0.175m first, and so on ad infinitum...so the arrow basically doesn't move and motion is not possible.

Many philosophers got crazy with these paradoxes. I don't like the math proofs too (it's like you are cheated twice). For me it's more like a mismatch in the limits of the mathematics to comprehend the universe. My second guess was that time is discrete and we cannot slice it ad infinitum.

Ok don't ask me to see if I know what I am saying...

[aitraaz]

My second guess was that time is discrete and we cannot slice it ad infinitum.

Nice, brings to mind the saying "Time is invention, or it is nothing at all"...

[ricardo]

I would take a simplistic approach and say that even if time is continuous, we are not, and the tools we have to measure both time and distance are discreet. Then, at some time, there will be a rounding error and Achilles will happen to pass the turtle, even if some theory tells otherwise.

So we are knee deep into miscalculations. Live with that.

Ricardo

[Frances]

I heard that the runner accidentally stepped on the tortoise while passing him. The tortoise made a full recovery though.

[Thomas An.]

Thomas, as far as I know the paradox is nonetheless mathematically solved with the introduction of calculus (leibniz)

Well, the common mathematical solutions (involving calculus) is a point where I would disagree.

Math does not really solve the problem. It only tells you that the point of "meeting" is a *limit* (an asymptotic boundary) and as such it falls right into the philosophical trap.

I remember, the issue of limits back in highschool and I was getting miffed everytime my roomates claimed they understood the lectures and could solve the exercises. Where in fact all we did was just "parrot" the methodology without really having a grasp of the underlying concepts.

Calculus is built on top of limits (an integral is a limit value of infinitescimal sums), but the concept of limit itself is a form of mathematical "duct-tape". We are looking at a series and observe a trend and see if a trend is kinda going towards a value. At that point we throw our arms in the air and say "screw it, we are not going to sit here all day thinking about it ... so lets just figure out which value it seems it wants to go to, but never seems to actually get at .... and then lets just use this *limiting* unattainable value in the rest of our calculations". Basically everywhere you see the word 'lim', you can replace it with the words "NeverSeemToAttain" (but this would be too long and impractical for daily use .... so we stick with 'lim')

1/2+1/4+1/8+1/16+..., which is a limit converging to 1 (not to infinite)

Exactly ! ... the math tells you that '1' is the limit (the unattainable boundary, the value it NeverSeemsToAttain)

In this case we could reduce the problem in a simple geometric series:
Point of meeting = 1000m+10m+0.1m + 0.001m + 0.00001m ..... =
=10^(3-0) + 10^(3-2) + 10^(3-4) .... + 10^(3-2n) =
=sum[n=1->inf] of 10^3 * (1/100)^(n-1) =
=10^3 / (1 - 1/100) =
=1010.101010101010101001...... meters

... but the solution to a geometric series involves limits .... a/(1-r) - a/(1-r)*lim r^n as n->inf

So the mathematical method does not really attack / solve the underlying issue ... it is basically telling us that the meeting point that they "NeverSeemToAttain" is at 100000/99 meters

... but the paradox already claims that there must be a point they never seem to attain. So the math offers nothing new here

Again, I would argue that the true fallacy in this paradox is that it requires infinite observer speed which would entail consumption of the universe in an attempt to pause time.

[JDHill]

Different notations/spaces/disciplines have different conceptual limitations...which is greater:

.333333333333333333333333 (...)

-or-

1/3

[Thomas An.]

...."Comprehend" is the wrong choice of word IMO. Is anything really infinite?....

ok, ... how many decimals are trully in this division ? --> 1/3
How many decimals does the constant 'pi' have ?

Lets say we shrink-in ourselves 1000000 trillion times .... now that becomes our new datum size (our reference point of measuring distances). Lets call this datum a1.

Now from datum a1 lets say we shrink ourselves 1000000 trillion times again and this new level becomes datum size a2

Now from datum a2 lets say we shrink ourselves 1000000 trillion times.

...
...
...
Now from datum an lets say we shrink ourselves 1000000 trillion times.
Now from datum a(n+1) lets say we shrink ourselves 1000000 trillion times.

where 'n' can be any number from 1 to ?? ... what ?

[Thomas An.]

Different notations/spaces/disciplines have different conceptual limitations...which is greater:

.333333333333333333333333 (...)

-or-

1/3

No, I was not refering to the notation issue two posts up (in regards to 1010.1010...).
The thing is that the fraction 10^3 / (1 - 1/100) is already a limit in itself. The proof of geometric series involves a limit causing one of the terms to drop out (having a limit at zero).

Now, even in regards to notation (since you brought it up):
a=0.3333...
10a=3.3333...
10a-a = 3
9a=3
a=3/9 --> a=1/3

In this case the assumption is that even though '10a' is shifted by an order of ten its decimal component is still somehow equal to 'a' .......

[superbad]

Maybe it's because I'm an engineer and not a mathturbator, but it seems a simple experiment could be setup to disprove the theory that Achilles will never catch the tortoise. You can even try it at home if you've got a tortoise handy.

[Thomas An.]

Maybe it's because I'm an engineer and not a mathturbator, but it seems a simple experiment could be setup to disprove the theory that Achilles will never catch the tortoise. You can even try it at home if you've got a tortoise handy.

The point is to beat the argument in its own game.
Since it operates on pure theoretical logic we are inclined to use pure theoretical logic in return to expose a fallacy.

[JDHill]

I wasn't disagreeing. In decimal, you will obviously never find the meeting point...but after 102 seconds, Achilles will have passed the tortise, disproving the contention. More accurately, they will meet in exactly 10000/99 seconds. The inability to express the solution in a decimal number doesn't negate its completeness. Any difficulty in pinpointing the exact point in time when they meet is not a failing of logic or mathematics...it is the nature of the infinite divisibility of the timeline. If one denies the fact that there are infinite possible frames between Achilles and the tortise, one will continue to check the next frame...and the next...and the next...ad infinitum.

...my 2cents.

[Thomas An.]

I wasn't disagreeing. In decimal, you will obviously never find the meeting point... More accurately, they will meet in exactly 10000/99 seconds...

Nope, again the argument is not about the decimal precision issue.

The argument is more along the lines that the 100000/99 value is the -->limit<--
(Try to review the earlier points a little more ... decimal precision was never claimed as an issue and as a matter of fact the formula 10^3/(1- (1/100)) was put in bold while its decimal value was not; in hope to avoid making it an issue )

Again, math is not really solving the matter ... it is falling right into it

Any difficulty in pinpointing the exact point in time when they meet is not a failing of logic or mathematics...

Well, the only pinpointing that math does is the *limit* of a point of meeting (as in a point where they seem to be assymptoticaly getting at, but not really attaining)

The paradox already claims that a meeting point would be assymptotic. The math basically confirms this (and puts a numeric value to it) but doesn't really attack the issue at heart.