Achilles and the Tortoise

Miscellaneous History
141

I think I solved the paradox. It appears that the confusion emerges from analyzing the problem at too-high a level and sliding a few mistakes in such as believing that the time intervals are constant.

NOTE: OPEN IMAGES IN A NEW WINDOW TO VIEW THEM IN PARALLEL WITH THE EXPLANATION.

Here is the problem graphically:

http://i.imgur.com/uHNe7.gif

At 1 you have your beginning situation. A is at 0 and B is at 8.

Say the speed of B is 2/sec and A is 8/sec.

  1. Now A has moved 8 units in a second to where B was before. B has moved only 2 units ahead.

  2. Now, B moves for a quarter of a second, so it goes 0.5 away, while A travels 2, to where B was before.

  3. B moves 0.125 while A moves 0.5 (in 1/16 of a second)

You may wonder why B is moving less and less each time. Well, for A to not have caught up to B yet, B can only move an amount of time in which A will not have had the chance to catch up with him.

What you see in the picture is that infinity is indeed involved. The problem is, not an infinity of time, but of divisions of space. To follow the conditions in the problem (ie, Achilles not catching up), we must make the time period of each motion smaller and smaller (this way Achilles can’t catch up). This means that eventually, at infinity time intervals, the two might converge. Thing is, add up all of these infinity of time intervals and you will get a constant time. What that means is that at a given, constant-valued time, A and B will converge after going through infinitely many intervals of time (yet still a finite sum).

Does the math support this answer? In fact, yes.

First, let’s look at the “naive” method of doing this, with which the paradox has a problem:

http://i.imgur.com/Wm6zE.gif

The position of B is as shown and A is as shown, both dependent on time t.

Setting them equal, we figure out that t=4/3 when they meet, and the distance from the origin is 32/8.

Now, the paradox says “that’s not true!”

Taking the insights I provided before with the first diagram (infinite time intervals, not infinite time), let’s see what series and calculus tell us:

http://i.imgur.com/N3TX0.gif

We start by figuring out where B will end up. It begins at 8, so that is the first term. Next, we have to add an infinite geometric series. The first term is 2, so 2 is the numerator. The ratio is 1/4, so 1-1/4 is in the denominator. Solving, we get 32/3.

For A, it starts at 0 and its first geometric term is 8 with ratio 1/4, evaluating, we have 32/3.

This provides support for the idea that the problem is in the infinite time intervals, not actually infinite time.

The solution might be made more rigorous if I used dt instead of t, but the idea is there.

Yes, by the time Achilles reaches the turtle’s starting point the turtle is ahead, but much closer than initially. This keeps going and becomes smaller and smaller, until they converge. But note, too, that for Achilles to be always behind, the time period needs to shorten every time. The paradox arises by having the reader think that the time period of the movements is the same. It’s not. It’s a geometric series with a ratio < 1, so it converges to a constant. It has infinite time intervals, but converges to a constant.

I feel proud.

142

I’m going to resurrect my syllogism because you’ve slyly dismissed it out-of-hand without dealing with the substance of its argument.

  1. PRQ is similar to TPS

  2. Therefore, the tan of PQR (= PR/PQ) and the tan of TPS (=TP/TS) are the same

  3. The base of TS is first divided into two segments - the segment marked ‘1’ and the remaining segments. Let’s name that point (directly below Q and R) U. Let’s name the point dividing the segment along TS marked r and the sgement marked r2 V

  4. TUR is similar to TPS. Therefore, the ratio SU:UT is the same as the ratio of UV:VT.

  5. Therefore, another similar triangle can be constructed that will be in the same ratio as the similar triangles TUR:TPS, ad infinitum - (this is the inductive step). Please show me how this point violates some fact about analysis or mathematical induction or any law of mathematics.

  6. Therefore, the proportion of PS:QR is equal to the infinite sum constructed along TS, that is, the ratio of the segment marked ‘r’ to the remaining segments through nesting similar triangles ad infinitum (from point 5): 1/(1-r) = 1 + r + r2 + …

Clayton -

143

Please show me how this point violates some fact about analysis or mathematical induction or any law of mathematics

It violates the accepted standard of mathematical proof. Mathematics is conducted within the framework of ordinary logical thinking. Much thought has been invested into what the rules of logical thinking are, which rules of deduction are valid, and which are invalid.

For example, say someone writes a proof that contains the following step:

“I count to three, clap my hands and make a wish. The tooth fairy then tells me that the answer is 5” QED.

This proof would not be accepted, because consulting the tooth fairy, though possibly totally reliable, has not been accepted by the mathematical community as valid.

The same is true of your step 5. Writing “ad infinitum” is exactly like consulting the tooth fairy, as far as validity of a proof is concerned. The mathematical community, after many disappointments, has rejected as inadmissable all proofs with the words “ad infinitum” in them, just as it has rejected proofs from the tooth fairy, and proofs based on pictures.

Now you may ask, but what about mathematical induction? Isn’t writing “ad infinitum” the very thing that one does when performing mathematical induction?

The answer is a resounding no. I wrote in an earlier post the very strict criteria required to correctly use mathematical induction. Step 5 fails those criteria decisively.

144

I also used the search on there. This is what I gathered from it: some formulas were offered, but this guy below broke down the point of the entire question. Historically, he’s correct, but they’ve squabbled with themselves over it too, just more akin to mathematically than we have.

I think the mathematical “explanation” of the Zeno’s paradox (convergence of infinite series) is quite unsatisfying. Assuming that each term in the series corresponds to one step of Achilles’s and considering that he indeed overtakes the turtle in finite time, which of Achilles feet is forward at the moment when he reaches the turtle?

Or a slightly different, but equivalent, presentation of the paradox: assume that the turtle changes direction at each discrete instant of time Achilles reaches her previous position, alternatively moving NE and SE. Achilles just follows her path. What direction is the turtle facing the moment Achilles reaches her?

Achilles’s and the turtle is no paradox at all, but a refutation of the hypotheses that the space is continuous. Zeno’s arrow paradox is a refutation of the hypothesis that the space is discrete. Together they form a paradox and an explanation is probably not easy. For Zeno the explanation was that what we perceive as motion is an illusion. In any case, I don’t think that convergent infinite series have anything to do with the heart of Zeno’s paradoxes.

EDIT: The same argument can be made point-like particles, only assuming that physical reality is continuous and infinitely divisible. Imagine a photon travelling between an infinite sequence of mirrors placed in a zig-zag shape with distance between mirrors decreasing at a geometric rate. So the photon bounces from a NE to SE direction and back, with the distance travelled decreasing “fast”. Since the length of the total path is finite (sum of a geometric series), the photon will emerge from the sequence of mirrors in finite time. What direction will it travel? The heart of the Zeno’s argument is that there is no logical way to decide that. You may argue that it is impossible to build such a sequence of mirrors, however this is just conceding Zeno’s point that physical reality is NOT continuous and infinitely divisible.

I think the mathematical model of the Zeno’s paradox is a great pedagogical tool in first year calculus, probably could be made even earlier in high school, but it misses an important aspect of Zeno’s argument. Granted, this argument lies at the boundary of math, physics and perhaps philosophy.

http://math.stackexchange.com/questions/142932/achilles-and-the-tortoise-paradox

145

This is brilliant. It is a response to Achilles vs. the Tortoise made by Dragon Lady on www.objectivismonline.com.

“…the entire nature of Zeno’s paradox means you’re treating mathematics as if they inform physics and not the other way around. Just because it’s possible to do something mathematically doesn’t mean it’s possible to do it physically.”

I think that sums it up nicely.

146

Objectivism is braindead.

147

PS:

has anyone read Godel Escher Bach?

148

Prove it.

149

Looks very interesting, Vive. I’ll check it out.

150

You just proved it for me.

Clayton -

151

Clayton, first, read this.

Now, please respond to my original request in an intelligent manner. You can do it, son.

152

Holy crap, how did you find that video of me???

:stuck_out_tongue:

Honestly, I really don’t care to prove or disprove anything about objectivism. I simply find it annoying and braindead. If I had to sum up in one sentence what I don’t like about it I suppose that it’s because it’s a kind of dogmatic nihilism which strikes me as an immediate contradiction.

I recommend you read everything about Epicurus you can get your hands on. Objectivism borrows a few good themes from Epicurean philosophy but then completely misses the whole point of life: your satisfaction.

Clayton -

153

Looks very interesting, Vive. I’ll check it out.

Yeah, if you like this paradox - it deals extensively with it you may like it. Been awhile since I read it though so I cant say much about it.

154

In addition to accepting the RHS as the least upper bound of the LHS, do you also accept that the LHS is exactly equal to the RHS, and they are in fact merely two ways of notating the same number, just as 1/2 and .5 are exactly the same number?

155

Jack,

An equality means equality, of course.

But we have to be careful with our definitions. The LHS has no other definition but the LUB of the infinite series .9+.09+…

An infinite series is not a number, and cannot equal a number. So that saying “the infinite series .9+.09+… equals 1” is false.

But the statement “the LUB of the infinite series .9+.09+… equals 1” is true.

And the equation means, by definition, the latter.

156

Sorry, no. An infinite series is a single number, by definition.

157

158

You’re using one or both of the words “equal” and “number” in a different way than mathematicians do. Please provide the cite from Hardy or anybody to back up this statement. Note that a least-upper-bound is a number and you’ve claimed multiple times in this thread that an infinite series equals its least-upper-bound.

Clayton -

159

I should clarify my comment. A series, by definition, is a single number. Adding infinite as a descriptor does not change that.

160

This isn’t quite true. An infinite divergent series is not a single number.